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of mathematical physics. I have been made aware 
of the importance of this problem by my friend and 
collaborator Dr. Whitehead, to whom are due almost 
all the differences between the views advocated here 
and those suggested in The Problems of Philosophy* 
I owe to hi the definition of points, the suggestion 
for the treatment of instants and " things," and the 
whole conception of the world of physics as a construc- 
tion rather than an inference. What is said on these 
topics here is, in fact, a rough preliminary account 
of the more precise results which he is giving in the 
fourth volume of our Principia Mathematical It will 
be seen that if his way of dealing with these topics is 
capable of being successfully carried through, a wholly 
new light is thrown on the time-honoured controversies 
of realists and idealists, and a method is obtained of 
solving all that is soluble in their problem. 

The speculations of the past as to the reality or 
unreality of the world of physics were baffled, at the 
outset, by the absence of any satisfactory theory of 
the mathematical infinite. This difficulty has been 
removed by the work of Georg Cantor. But the 
positive and detailed solution of the problem by means 
of mathematical constructions based upon sensible 
objects as data has only been rendered possible by the 
growth of mathematical logic, without which it is 
practically impossible to manipulate ideas of the 
requisite abstractness and complexity. This aspect, 
which is somewhat obscured in a merely popular 
outline such as is contained in the following lectures, 
will become plain as soon as Dr. Whitehead's work is 
published. In pure logic, which, however, will be very 

LondonandNewYork, 1912 ("Home University Library"), 
a The first volume was published at Cambridge in 1910, the 
second in 1912, and the third in 1913. 


briefly discussed in these lectures, I have had the 
benefit of vitally important discoveries, not yet pub- 
lished, by my friend Mr. Ludwig Wittgenstein. 

Since my purpose was to illustrate method, I have 
included much that is tentative and incomplete, for 
it is not by the study of finished structures alone that 
the manner of construction can be learnt. Except 
in regard to such matters as Cantor's theory of infinity, 
no finality is claimed for the theories suggested ; but 
I believe that where they are found to require modi- 
fication, this will be discovered by substantially the 
same method as that which at present makes them 
appear probable, and it is on this ground that I ask the 
reader to be tolerant of their incompleteness. 

June 1914. 



I. Current Tendencies 13 

II. Logic as the Essence of Philosophy 42 

III. On Our Knowledge of the External World , 70 

IV. The World of Physics and the World of Sense 106 
V. The Theory of Continuity 135 

VI. The Problem of Infinity Considered Historically 159 

VII. The Positive Theory of Infinity 189 

VIII. On the Notion of Cause, with Applications to 

the Free-Will Problem 214 

Index 247 



PHILOSOPHY, from the earliest times, has made 
greater ml aim a, and achieved fewer results, than 
any other branch, of learning. Ever since Thales 
said that all is water, philosophers have been ready 
with glib assertions about the sum-total of things ; 
and equally glib denials have come from other philo- 
sophers ever since Thales was contradicted by Anaxi- 
mander. I believe that the time has now arrived 
when this unsatisfactory state of things can be brought 
to an end. In the following course of lectures I 
shall try, chiefly by taking certain special problems as 
examples, to indicate wherein the claims of philo- 
sophers have been excessive, and why their achieve- 
ments have not been greater. The problems and the 
method of philosophy have, I believe, been miscon- 
ceived by all schools, many of its traditional problems 
being insoluble with our means of knowledge, while 
other more neglected but not less important problems 
can, by a more patient and more adequate method, be 
solved with all the precision and certainty to which the 
most advanced sciences have attained. 

Among present-day philosophies, we may distin- 


guish three principal types, often combined in varying 
proportions by a single philosopher, but in essence 
and tendency distinct. The first of these, which I 
shall call the classical tradition, descends in the main 
from Kant and Hegel ; it represents the attempt to 
adapt to present needs the methods and results of 
the great constructive philosophers from Plato down- 
wards. The second type, which may be called evolu- 
tionism, derived its predominance from Darwin, 
and must be reckoned as having had Herbert Spencer 
for its first philosophical representative ; but in 
recent times it has become, chiefly through William 
James and M. Bergson, far bolder and far more search- 
ing in its innovations than it was in the hands of 
Herbert Spencer. The third type, which may be called 
" logical atomism " for want of a better name, has 
gradually crept into philosophy through the critical 
scrutiny of mathematics. This type of philosophy, 
which is the one that I wish to advocate, has not as 
yet many whole-hearted adherents, but the "new 
realism" which owes its inception to Harvard is 
very largely impregnated with its spirit. It repre- 
sents, I believe, the same kind of advance as was 
introduced into physics by Galileo : the substitution of 
piecemeal, detailed, and verifiable results for large 
untested generalities recommended only by a certain 
appeal to imagination. But before we can understand 
the changes advocated by this new philosophy, we 
must briefly examine and criticize the other two types 
with which it has to contend. 


Twenty years ago, the classical tradition, having 
vanquished the opposing tradition of the English 


empiricists, held almost unquestioned sway in all 
Anglo-Saxon universities. At the present day, though 
it is losing ground, many of the most prominent 
teachers still adhere to it. In academic France, in 
spite of M. Bergson, it is far stronger than all its 
opponents combined ; and in Germany it had many 
vigorous advocates. Nevertheless, it represents on 
the whole a decaying force, and it has failed to adapt 
itself to the temper of the age. Its advocates are, 
in the main, those whose extra-philosophical know- 
ledge is liteniry, rather than those who have felt the 
inspiration of science. There are, apart from reasoned 
arguments, certain general intellectual forces against 
it the same general forces which are breaking down 
the other great syntheses of the past, and making our 
age one of bewildered grouping where our ancestors 
walked in the clear daylight of unquestioning certainty. 
The original impulse out of which the classical 
tradition developed was the naive faith of the Greek 
philosophers in the omnipotence of reasoning. The 
discovery of geometry had intoxicated them, and its 
a priori deductive method appeared capable of universal 
application. They would prove, for instance, that 
all reality is one, that there is no such thing as change, 
that the world of sense is a world of mere illusion ; 
and the strangeness of their results gave them no 
qualms because they believed in the correctness of 
their reasoning. Thus it came to be thought that 
by mere thinking the most surprising and important 
truths concerning the whole of reality could be estab- 
lished with a certainty which no contrary observations 
could shake. As the vital impulse of the early philo- 
sophers died away, its place was taken by authority 
and tradition, reinforced, in the Middle Ages and 
almost to our own day, by systematic theology. 


Modern philosophy, from Descartes onwards, though 
not bound by authority like that of the Middle Ages, 
still accepted more or less uncritically the Aristotelian 
logic. Moreover, it still believed, except in Great 
Britain, that a priori reasoning could reveal otherwise 
undiscoverable secrets about the universe, and could 
prove reality to be quite different from what, to direct 
observation, it appears to be. It is this belief, rather 
than any particular tenets resulting from it, that I 
regard as the distinguishing characteristic of the 
classical tradition, and as hitherto the main obstacle 
to a scientific attitude in philosophy. 

The nature of the philosophy embodied in the 
classical tradition may be made clearer by taking a 
particular exponent as an illustration. For this 
purpose, let us consider for a moment the doctrines 
of Mr. Bradley, who is probably the most distinguished 
British representative of this school. Mr. Bradley's 
Appearance and Reality is a book consisting of two 
parts, the first called Appearance, the second Reality. 
The first part examines and condemns almost all that 
makes up our everyday world : things and qualities, 
relations, space and time, change, causation, activity, 
the self. All these, though in some sense facts which 
qualify reality, are not real as they appear. What 
is real is one single, indivisible, timeless whole, called 
the Absolute, which is in some sense spiritual, but does 
not consist of souls, or of thought and will as we know 
them. And all this is established by abstract logical 
reasoning professing to find self-contradictions in the 
categories condemned as mere appearance, and to 
leave no tenable alternative to the kind of Absolute 
which is finally affirmed to be real. 

One brief example may suffice to illustrate Mr. Brad- 
ley's method. The world appears to be full of 


many things with various relations to each other 
right and left, before and after, father and son, and 
so on. But relations, according to Mr. Bradley, axe 
found on examination to be self-contradictory and 
therefore impossible. He first argues that, if there 
are relations, there must be qualities between which 
they hold. This part of his argument need not detain 
us. He then proceeds : 

"But how the relation can stand to the qualities 
is, on the other side, unintelligible. If it is nothing 
to the qualities, then they are not related at all; 
and, if so, as we saw, they have ceased to be qualities, 
and their relation is a nonentity. But if it is to be 
something to them, then clearly we shall require a 
new connecting relation. For the relation hardly 
can be the mere adjective of one or both of its terms ; 
or, at least, as such it seems indefensible. And, 
being something itself, if it does not itself bear a rela- 
tion to the terms, in what intelligible way will it 
succeed in being anything to them ? But here again 
we are hurried off into the eddy of a hopeless process, 
since we are forced to go on finding new relations 
without end. The links are united by a link, and this 
bond of union is a link which also has two ends ; and 
these require each a fresh IfriTr to connect them with 
the old. The problem is to find how the relation 
can stand to its qualities, and this problem is 
insoluble." I 

I do not propose to examine this argument in 

detail, or to show the exact points where, in my opinion, 

it is fallacious. I have quoted it only as an example 

of method. Most people will admit, I think, that it 

is calculated to produce bewilderment rather than 

conviction, because there is more likelihood of error 

1 Appearance and Reality, pp. 32-33. 



in a very subtle, abstract, and difficult argument 
than in so patent a fact as the interrelatedness of the 
things in the world. To the early Greeks, to whom 
geometry was practically the only known science, it. 
was possible to follow reasoning with assent even 
when it led to the strangest conclusions. But to us, 
with our methods of experiment and observation, our 
knowledge of the long history of a priori errors refuted 
by empirical science, it has become natural to suspect 
a fallacy in any deduction of which the conclusion 
appears to contradict patent facts. It is easy to 
carry such suspicion too far, and it is very desirable, if 
possible, actually to discover the exact nature of the 
error when it exists. But there is no doubt that what 
we may call the empirical outlook has become part 
of most educated people's habit of mind ; and it is 
this, rather than any definite argument, that has 
diminished the hold of the classical tradition upon 
students of philosophy and the instructed public 

The function of logic in philosophy, as I shall try 
to show at a later stage, is all-important ; but I do 
not think its function is that which it has in the classi- 
cal tradition. In that tradition, logic becomes con- 
structive through negation. Where a number of 
alternatives seem, at first sight, to be equally possible, 
logic is made to condemn all of them except one, 
and that one is then pronounced to be realized in the 
actual world. Thus the world is constructed by means 
of logic, with little or no appeal to concrete experience. 
The true function of logic is, in my opinion, exactly 
the opposite of this. As applied to matters of experi- 
ence, it is analytic rather than constructive ; taken 
a priori, it shows the possibility of hitherto unsus- 
pected alternatives more often than the impossibility 


of alternatives which seemed prima fade possible. 
Thus, while it liberates imagination as to what the 
world may be, it refuses to legislate as to what the 
world is. This change, which has been brought about 
by an internal revolution in logic, has swept away the 
ambitious constructions of traditional metaphysics, 
even for those whose faith in logic is greatest ; while 
to the many who regard logic as a chimera the para- 
doxical systems to which it has given rise do not seem 
worthy even of refutation. Thus on all sides these 
systems have ceased to attract, and even the philo- 
sophical world tends more and more to pass them by. 

One or two of the favourite doctrines of the school 
we are considering may be mentioned to illustrate 
the nature of its claims. The universe, it tells us, is 
an " organic unity," like an animal or a perfect work 
of art. By this it means, roughly speaking, that all 
the different parts fit together and co-operate, and 
are what they are because of their place in the whole. 
This belief is sometimes advanced dogmatically, 
while at other times it is defended by certain logical 
arguments, If it is true, every part of the universe 
is a microcosm, a miniature reflection of the whole. 
If we knew ourselves thoroughly, according to this 
doctrine, we should know everything. Common sense 
would naturally object that there are people say 
in China with whom our relations are so indirect 
and trivial that we cannot infer anything important 
as to them from any fact about ourselves. If there 
are living beings in Mars or in more distant parts of 
the universe, the same argument becomes even stronger. 
But further, perhaps the whole contents of the space 
and time in which we live form only one of many 
universes, each seeming to itself complete. And thus 
the conception of the necessary unity of all that is 


resolves itself into the poverty of imagination, and a 
freer logic emancipates us from the strait-waistcoated 
benevolent institution which idealism palms off as the 
totality of being. 

Another very important doctrine held by most, 
though not all, of the school we are examining is the 
doctrine that all reality is what is called " mental " 
or "spiritual," or that, at any rate, all reality is 
dependent for its existence upon what is mental. 
This view is often particularized into the form which 
states that the relation of knower and known is funda- 
mental, and that nothing can exist unless it either 
knows or is known. Here again the same legislative 
function is ascribed to a priori argumentation : it 
is thought that there are contradictions in an unknown 
reality. Again, if I am not mistaken, the argument 
is fallacious, and a better logic will show that no limits 
can be set to the extent and nature of the unknown. 
And when I speak of the unknown, I do not mean 
merely what we personally do not know, but what 
is not known to any mind. Here as elsewhere, while 
the older logic shut out possibilities and imprisoned 
imagination within the walls of the familiar, the newer 
logic shows rather what may happen, and refuses to 
decide as to what must happen. 

The classical tradition in philosophy is the last sur- 
viving child of two very diverse parents : the Greek 
belief in reason, and the mediaeval belief in the tidi- 
ness of the universe. To the schoolmen, who lived 
amid wars, massacres, and pestilences, nothing 
appeared so delightful as safety and order. In their 
idealising dreams, it was safety and order that they 
sought : the universe of Thomas Aquinas or Dante is 
as small and neat as a Dutch interior. To us, to 
whom safety has become monotony, to whom the 


primeval savageries of nature are so remote as to 
become a mere pleasing condiment to our ordered 
routine, the world of dreams is very different from what 
it was amid the wars of Guelf and Ghibelline. Hence 
William James's protest against what he calls the 
" block universe " of the classical tradition ; hence 
Nietzsche's worship of force ; hence the verbal blood- ' 
thirstiness of many quiet literary men. The barbaric 
substratum of human nature, unsatisfied in action, 
finds an outlet in imagination. In philosophy, as 
elsewhere, this tendency is visible ; and it is this, rather 
than formal argument, that has thrust aside the classical 
tradition for a philosophy which fancies itself more 
virile and more vital. 1 


Evolutionism, in one form or another, is the pre- 
vailing creed of our time. It dominates our politics, 
our literature, and not least our philosophy. Nietzsche, 
pragmatism, Bergson, are phases in its philosophic 
development, and their popularity far beyond the 
circles of professional philosophers shows its conso- 
nance with the spirit of the age. It believes itself 
firmly based on science, a liberator of hopes, an iospirer 
of an invigorating faith in human power, a sure anti- 
dote to the ratiocinative authority of the Greeks and 
the dogmatic authority of mediaeval systems. Against 
so fashionable and so agreeable a creed it may seem 
useless to raise a protest ; and with much of its spirit 
every modern man must be in sympathy. But I 
think that, in the intoxication of a quick success, much 
that is important, and vital to a true understanding of 
the universe has been forgotten. Something of Hellenism 
1 Written before August 1914. 


must be combined with the new spirit before it can 
emerge from the ardour of youth into the wisdom of 
manhood. And it is time to remember that biology 
is neither the only science, nor yet the model to which 
all other sciences must adapt themselves. Evolu- 
tionism, as I shall try to show, is not a truly scientific 
philosophy, either in its method or in the problems 
which it considers. The true scientific philosophy 
is something more arduous and more aloof, appealing 
to less mundane hopes, and requiring a severer dis- 
cipline for its successful practice. 

Darwin's Origin of Species persuaded the world 
that the difference between different species of animals 
and plants is not the fixed, immutable difference 
that it appears to be. The doctrine of natural kinds, 
which had rendered classification easy and definite, 
which was enshrined in the Aristotelian tradition, 
and protected by its supposed necessity for orthodox 
dogma, was suddenly swept away for ever out of the 
biological world. The difference between man and 
the lower animals, which to our human conceit appears 
enormous, was shown to be a gradual achievement, 
involving intermediate beings who could not with 
certainty be placed either within or without the human 
family. The sun and planets had already been shown 
by Laplace to be very probably derived from a primi- 
tive more or less undifferentiated nebula. Thus the 
old fixed landmarks became wavering and indistinct, 
and all sharp outlines were blurred. Things and 
species lost their boundaries, and none could say where 
they began or where they ended. 

But if human conceit was staggered for a moment 
by its kinship with the ape, it soon found a way to 
reassert itself, and that way is the " philosophy " of 
evolution. A process which led from the amoeba to 


man appeared to the philosophers to be obviously a 
progress though whether the amoeba would agree 
with this opinion is not known. Hence the cycle of 
changes which science had shown to be the probable 
history of the past was welcomed as revealing a law 
of development towards good in the universe an 
evolution or unfolding of an ideal slowly embodying 
itself in the actual. But such a view, though it might 
satisfy Spencer and those whom we may call Hegelian 
evolutionists, could not be accepted as adequate by 
the more whole-hearted votaries of change. An ideal 
to which the world continuously approaches is, to 
these minds, too dead and static to be inspiring. Not 
only the aspirations, but the ideal too, must change 
and develop with the course of evolution ; there must 
be no fixed goal, but a continual fashioning of fresh 
needs by the impulse which is life and which alone 
gives unity to the process. 

Ever since the seventeenth century, those whom 
William James described as the " tender-minded " 
have been engaged in a desperate struggle with the 
mechanical view of the course of nature which physical 
science seems to impose. A great part of the attractive- 
ness of the classical tradition was due to the partial 
escape from mechanism which it provided. But now, 
with the influence of biology, the " tender-minded " 
believe that a more radical escape is possible, sweeping 
aside not merely the laws of physics, but the whole 
apparently immutable apparatus of logic, with its 
fixed concepts, its general principles, and its reasonings 
which seem able to compel even the most unwilling 
assent. The older kind of teleology, therefore, which 
regarded the End as a fixed goal, already partially 
visible, towards which we were gradually approaching, 
is rejected by M. Bergson as not allowing enough for 


the absolute dominion of change. After explaining 
why he does not accept mechanism, he proceeds : t 

" But radical finalism is quite as unacceptable, and 
for the same reason. The doctrine of teleology, in 
its extreme form, as we find it in Leibniz for example, 
implies that things and beings merely realize a pro- 
gramme previously arranged. But if there is nothing 
unforeseen, no invention or creation in the universe, 
time is useless again. As in the mechanistic hypo- 
thesis, here again it is supposed that all is given. 
Finalism thus understood is only inverted mechanism. 
It springs from the same postulate, with this sole 
difference, that in the movement of our finite intellects 
along successive things, whose successiveness is reduced 
to a mere appearance, it holds in front of us the light 
with which it claims to guide us, instead of putting it 
behind. It substitutes the attraction of the future 
for the impulsion of the past. But succession remains 
none the less a mere appearance, as indeed does 
movement itself. In the doctrine of Leibniz, time is 
reduced to a confused perception, relative to the human 
standpoint, a perception which would vanish, like a 
rising mist, for a mind seated at the centre of things. 

" Yet finalism is not, like mechanism, a doctrine 
with fixed rigid outlines. It admits of as many inflec- 
tions as we like. The mechanistic philosophy is to 
be taken or left : it must be left if the least grain of 
dust, by straying from the path foreseen by mechanics, 
should show the slightest trace of spontaneity. The 
doctrine of final causes, on the contrary, will never 
be definitively refuted. If one form of it be put aside, 
it will take another. Its principle, which is essentially 
psychological, is very flexible. It is so extensible, 
and thereby so comprehensive, that one accepts some- 
* Creative Evolution, Tfrigijgh translation, p. 41. 


thing of it as soon as one rejects pure mechanism. 
The theory we shall put forward in this book will 
therefore necessarily partake of finalism to a certain 

M. Bergson's form of finalism depends upon his 
conception of life. Life, in his philosophy, is a con- 
tinuous stream, in which all divisions are artificial 
and unreal. Separate things, beginnings and endings, 
are mere convenient fictions : there is only smooth, 
unbroken transition. The beliefs of to-day may count 
as true to-day, if they cany us along the stream ; but 
to-morrow they will be false, and must be replaced 
by new beliefs to meet the new situation. All our 
thinking consists of convenient fictions, imaginary 
congealings of the stream : reality flows on in spite 
of all our fictions, and though it can be lived, it cannot 
be conceived in thought. Somehow, without explicit 
statement, the assurance is slipped in that the future, 
though we cannot foresee it, will be better than the 
past or the present : the reader is like the child who 
expects a sweet because it has been told to open its 
mouth and shut its eyes. Logic, mathematics, physics, 
disappear in this philosophy, because they are too 
" static " ; what is real is an impulse and movement 
towards a goal which, like the rainbow, recedes as we 
advance, and makes every place different when we 
reach it from what it appeared to be at a distance. 

Now I do not propose at present to enter upon a 
technical examination of this philosophy. At present 
I wish to make only two criticisms of it first, that 
its truth does not follow from what science has ren- 
dered probable concerning the facts of evolution, and 
secondly, that the motives and interests which inspire 
it are so exclusively practical, and the problems with 
which it deals are so special, that it can hardly be 


regarded as really touching any of the questions that 
to my mind constitute genuine philosophy. 

(i) What biology has rendered probable is that the 
diverse species arose by adaptation from a less differen- 
tiated ancestry. This fact is in itself exceedingly 
interesting, but it is not the kind of fact from which 
philosophical consequences follow. Philosophy is 
general, and takes an impartial interest in all that 
exists. The changes suffered by minute portions of 
matter on the earth's surface are very important to us 
as active sentient beings ; but to us as philosophers 
they have no greater interest than other changes in 
portions of matter elsewhere. And if the changes on 
the earth's surface during the last few minions of years 
appear to our present ethical notions to be in the 
nature of a progress, that gives no ground for believing 
that progress is a general law of the universe. Except 
under the influence of desire, no one would admit for 
a moment so crude a generalization from such a tiny 
selection of facts. What does result, not specially 
from biology, but from all the sciences which deal with 
what exists, is that we cannot understand the world 
unless we can understand change and continuity. 
This is even more evident in physics than it is in 
biology. But the analysis of change and continuity is 
not a problem upon which either physics or biology 
throws any light: it is a problem of a new kind, 
belonging to a different kind of study. The question 
whether evolutionism offers a true or a false answer to 
this problem is not, therefore, a question to be solved 
by appeals to particular facts, such as biology and 
physics reveal. In assuming dogmatically a certain 
answer to this question, evolutionism ceases to be 
scientific, yet it is only in touching on this question 
that evolutionism reaches the subject-matter of philo 


sophy. Evolutionism thus consists of two parts : 
one not philosophical, but only a hasty generalization 
of the kind which the special sciences might hereafter 
confirm or confute; the other not scientific, but a 
mere unsupported dogma, belonging to philosophy by 
its subject-matter, but in no way deducible from the 
facts upon which evolutionism relies. 

(2) The predominant interest of evolutionism is in 
the question of human destiny, or at least of the 
destiny of life. It is more interested in morality 
and happiness than in knowledge for its own sake. 
It must be admitted that the same may be said of 
many other philosophies, and that a desire for the 
kind of knowledge which philosophy really can give 
is very rare. But if philosophy is to become scientific 
and it is our object to discover how this can be 
achieved it is necessary first and foremost that philo- 
sophers should acquire the disinterested intellectual 
curiosity which characterizes the genuine man of 
science. Knowledge concerning the future which is 
the kind of knowledge that must be sought if we are 
to know about human destiny is possible within 
certain narrow limits. It is impossible to say how 
much the limits may be 'enlarged with the progress 
of science. But what is evident is that any proposi- 
tion about the future belongs by its subject-matter 
to some particular science, and is to be ascertained, 
if at all, by the methods of that science. Philosophy 
is not a short cut to the same kind of results as those 
of the other sciences : if it is to be a genuine study, 
it must have a province of its own, and aim at results 
which the other sciences can neither prove nor disprove. 

The consideration that philosophy, if there is such 
a study, must consist of propositions which could 
not occur in the other sciences, is one which has very 


far-reaching consequences. All the questions which 
have what is called a human interest such, for 
example, as the question of a future life belong, at 
least in theory, to special sciences, and are capable, 
at least in theory, of being decided by empirical 
evidence. Philosophers have too often, in the past, 
permitted themselves to pronounce on empirical 
questions, and found themselves, as a result, in dis- 
astrous conflict with well-attested facts. We must, 
therefore, renounce the hope that philosophy can 
promise satisfaction to our mundane desires. What 
it can do, when it is purified from all practical taint, 
is to help us to understand the general aspects of the 
world and the logical analysis of familiar but complex 
things. Through this achievement, by the suggestion 
of fruitful hypotheses, it may be indirectly useful in 
other sciences, notably mathematics, physics, and 
psychology. But a genuinely scientific philosophy 
cannot hope to appeal to any except those who have 
the wish to understand, to escape from intellectual 
bewilderment. It offers, in its own domain, the kind 
of satisfaction which the other sciences offer. But 
it does not offer, or attempt to offer, a solution of the 
problem of human destiny, or of the destiny of the 

Evolutionism, if what has been said is true, is to be 
regarded as a hasty generalization from certain rather 
special facts, accompanied by a dogmatic rejection 
of all attempts at analysis, and inspired by interests 
which are practical rather than theoretical. In spite, 
therefore, of its appeal to detailed results in various 
sciences, it cannot be regarded as any more*genuinely 
scientific than the classical tradition which it has 
replaced. How philosophy is to be rendered scientific, 
and what is the true subject-matter of philosophy, 


I shall try to show first by examples of certain achieved 
results, and then more generally. We will begin 
with the problem of the physical conceptions of space 
and time and matter, which, as we have seen, are 
challenged by the contentions of the evolutionists. 
That these conceptions stand in need of reconstruc- 
tion will be admitted, and is indeed increasingly urged 
by physicists themselves. It will also be admitted that 
the reconstruction must take more account of change 
and the universal flux than is done in the older 
mechanics with its fundamental conception of an 
indestructible matter. But I do not think the recon- 
struction required is on Bergsonian lines, nor do I 
think that his rejection of logic can be anything but 
harmful. I shall not, however, adopt the method 
of explicit controversy, but rather the method of 
independent inquiry, starting from what, in a pre- 
philosophic stage, appear to be facts, and keeping 
always as dose to these initial data as the requirements 
of consistency will permit. 

Although explicit controversy is almost always 
fruitless in philosophy, owing to the fact that no two 
philosophers ever understand one another, yet it 
seems necessary to say something at the outset in 
justification of the scientific as against the mystical 
attitude. Metaphysics, from the first, has been 
developed by the union or the conflict of these two 
attitudes. Among the earliest Greek philosophers, 
the lonians were more scientific, and the Sicilians more 
mystical. 1 But among the latter, Pythagoras, for 
example, was in himself a curious mixture of the two 
tendencies: the scientific attitude led him to his 
proposition on right-angled triangles, while his mystic 
insight showed him that it is wicked to eat beans. 
* Cf. Burnet, Early Greek Philosophy, pp. 85 ff. 


Naturally enough, his followers divided into two 
sects, the lovers of right-angled triangles and the 
abhorrers of beans ; but the former sect died out, 
leaving, however, a haunting flavour of mysticism 
over much Greek mathematical speculation, and in 
particular over Plato's views on mathematics. Plato, 
of course, embodies both the scientific and mystical 
attitudes in a higher form than his predecessors, 
but the mystical attitude is distinctly the stronger 
of the two, and secures ultimate victory whenever 
the conflict is sharp. Plato, moreover, adopted from 
the Eleatics the device of using logic to defeat common 
sense, and thus to leave the field clear for mysticism 
a device still employed in our own day by the adherents 
of the classical tradition. 

The logic used in defence of mysticism seems to me 
faulty as logic, and in a later lecture I shall criticize 
it on this ground. But the more thoroughgoing 
mystics do not employ logic, which they despise: 
they appeal instead directly to the immediate deliver- 
ance of their insight. Now, although fully developed 
mysticism is rare in the West, some tincture of it 
colours the thoughts of many people, particularly as 
regards matter on which they have strong convictions 
not based on evidence. In all who seek passionately 
for the fugitive and difficult goods, the conviction 
is almost irresistible that there is in the world some- 
thing deeper, more significant, than the multiplicity 
of little facts chronicled and classified by science. 
Behind the veil of these mundane things, they feel 
something quite different obscurely shimmers, shining 
forth clearly in the great moments of illumination, 
which alone give anything worthy to be called real 
knowledge of truth. To seek such moments, therefore, 
is to them the way of wisdom, rather than, like the 


man of science, to observe coolly, to analyse without 
emotion, and to accept without question the equal 
reality of the trivial and the important. 

Of the reality or unreality of the mystic's world I 
know nothing. I have no wish to deny it, nor even 
to declare that the insight which reveals it is not a 
genuine insight. What I do wish to maintain 
and it is here that the scientific attitude becomes 
imperative is that insight, untested and unsupported, 
is an insufficient guarantee of truth, in spite of the fact 
that much of the most important truth is first sug- 
gested by its means. It is common to speak of an 
opposition between instinct and reason; in the 
eighteenth century, the opposition was drawn in 
favour of reason, but under the influence of Rousseau 
and the romantic movement instinct was given the 
preference, first by those who rebelled against arti- 
ficial forms of government and thought, and then, 
as the purely rationalistic defence of traditional theo- 
logy became increasingly difficult, by all who felt in 
science a menace to creeds which they associated 
with a spiritual outlook on life and the world. Berg- 
son, under the name of " intuition," has raised instinct 
to the position of sole arbiter of metaphysical truth. 
But in fact the opposition of instinct and reason is 
mainly illusory. Instinct, intuition, or insight is 
what first leads to the beliefs which subsequent reason 
confirms or confutes ; but the confirmation, where it 
is possible, consists, in the last analysis, of agreement 
with other beliefs no less instinctive. Reason is a 
harmonizing, controlling force rather than a creative 
one. Even in the most purely logical realms, it is 
insight that first arrives at what is new. 

Where instinct and reason do sometimes conflict 
is in regard to single beliefs, held instinctively, and 


held with such determination that no degree of incon- 
sistency with other beliefs leads to their abandon- 
ment. Instinct, like all human faculties, is liable to 
error Those in whom reason is weak are often un- 
willing to admit this as regards themselves, though 
all admit it in regard to others. Where instinct is 
least liable to error is in practical matters as to which 
right judgment is a help to survival; friendship 
and hostility in others, for instance, are often felt 
with extraordinary discrimination through very care- 
ful disguises. But even in such matters a wrong im- 
pression may be given by reserve or flattery ; and 
in matters less directly practical, such as philosophy 
deals with, very strong instinctive beliefs may be 
wholly mistaken, as we may come to know through 
their perceived inconsistency with other equally 
strong beliefs. It is such considerations that necessi- 
tate the harmonizing mediation of reason, which 
tests our beliefs by their mutual compatibility, and 
examines, in doubtful cases, the possible sources of 
error on the one side and on the other. In this there 
is no opposition to instinct as a whole, but only to 
blind reliance upon some one interesting aspect of 
instinct to the exclusion of other more commonplace 
but not less trustworthy aspects. It is such one- 
sidedness, not instinct itself, that reason aims at 

These more or less trite maxims may be illustrated 
by application to Bergson's advocacy of " intuition " 
as against "intellect." There are, he says, "two 
profoundly different ways of knowing a thing. The 
first implies that we move round the object ; the 
second that we enter into it. The first depends on 
the point of view at which we are placed and on the 
symbols by which we express ourselves. The second 


neither depends on a point of view nor relies on any 
symbol. The first kind of knowledge may be said 
to stop at the relative ; the second, in those cases 
where it is possible, to attain the absolute" z The 
second of these, which is intuition, is, he says, " the 
kind of intellectual sympathy by which one places 
oneself within an object in order to coincide with 
what is unique in it and therefore inexpressible" 
(p. 6). In illustration, he mentions self-knowledge : 
" there is one reality, at least, which we all seize from 
within, by intuition and not by simple analysis. It 
is our own personality in its flowing through time 
our self which endures " (p. 8). The rest of Bergson's 
philosophy consists in reporting, through the imper- 
fect medium of words, the knowledge gained by intui- 
tion, and the consequent complete condemnation of 
all the pretended knowledge derived from science and 
common sense. 

This procedure, since it takes sides in a conflict of 
instinctive beliefs, stands in need of justification by 
proving the greater trustworthiness of the beliefs on 
one side than of those on the other. Bergson attempts 
this justification in two ways first, by explaining that 
intellect is a purely practical faculty designed to secure 
biological success ; secondly, by mentioning remark- 
able feats of instinct in animals, and by pointing out 
characteristics of the world which, though intuition 
can apprehend them, axe baffling to intellect as he 
interprets it. 

Of Bergson's theory that intellect is a purely prac- 
tical faculty developed in the struggle for survival, 
and not a source of true beliefs, we may say, first, that 
it is only through intellect that we know of the struggle 
for survival and of the biological ancestry of man : if 
i Introduction to Metaphysics, p. i. 


the intellect is misleading, the whole of this merely 
inferred history is presumably untrue. If, on the 
other hand, we agree with M. Bergson in thinking that 
evolution took place as Darwin believed, then it is 
not only intellect, but all our faculties, that have been 
developed under the stress of practical utility. In- 
tuition is seen at its best where it is directly useful 
for example, in regard to other people's characters 
and dispositions. Bergson apparently holds that 
capacity for this kind of knowledge is less explicable 
by the struggle for existence than, for example, 
capacity for pure mathematics. Yet the savage 
deceived by false friendship is likely to pay for his 
mistake with his life ; whereas even in the most 
civilized societies men are not put to death for mathe- 
matical incompetence. All the most striking of his 
instances of intuition in animals have a very direct 
survival value. The fact is, of course, that both in- 
tuition and intellect have been developed because 
they axe useful, and that, speaking broadly, they are 
useful when they give truth and become harmful 
when they give falsehood. Intellect, in civilized man, 
like artistic capacity, has occasionally been developed 
beyond the point where it is useful to the individual ; 
intuition, on the other hand, seems on the whole to 
diminish as civilization increases. Speaking broadly, 
it is greater in children than in adults, in the un- 
educated than in the educated. Probably in dogs it 
exceeds anything to be found in human beings. But 
those who find in these facts a recommendation of 
intuition ought to return to running wild in the woods, 
dyeing themselves with woad and living on hips and 

Let us next examine whether intuition possesses any 
such infaJlibiJity as Bergson claims for it. The best 


instance of it, according to him, is our acquaintance 
with ourselves; yet self-knowledge is proverbially 
rare and difficult. Most men, for example, have in 
their nature meannesses, vanities, -and envies of which 
they are quite unconscious, though even their best 
friends can perceive them without any difficulty. It 
is true that intuition has a convincingness which is 
lacking to intellect : while it is present, it is almost 
impossible to doubt its truth. But if it should appear, 
on examination, to be at least as fallible as intellect, 
its greater subjective certainty becomes a demerit, 
making it only the more irresistibly deceptive. Apart 
from self-knowledge, one of the most notable examples 
of intuition is the knowledge people believe themselves 
to possess of those with whom they are in love : the 
wall between different personalities seems to become 
transparent, and people flifriTr they see into another 
soul as into their own. Yet deception in such cases 
is constantly practised with success ; and even where 
there is no intentional deception, experience gradually 
proves, as a rule, that the supposed insight was illusory, 
and that the slower, more groping methods of the 
intellect are in the long run more reliable. 

Bergson maintains that intellect can only deal 
with things in so far as they resemble what has been 
experienced in the past, while intuition has the power 
of apprehending the uniqueness and novelty that 
always belong to each fresh moment. That there is 
something unique and new at every moment, is cer- 
tainly true ; it is also true that this cannot be fully 
expressed by means of intellectual concepts. Only 
direct acquaintance can give knowledge of what is 
unique and new. But direct acquaintance of this 
kind is given fully in sensation, and does not require, 
so far as I can see, any special faculty of intuition for 


its apprehension. It is neither intellect nor intuition, 
but sensation, that supplies new data ; but when the 
data are new in any remarkable manner, intellect is 
much more capable of dealing with them than intui- 
tion would be. The hen with a brood of ducklings 
no doubt has intuitions which seem to place her inside 
them, and not merely to know them analytically ; 
but when the ducklings take to the water, the whole 
apparent intuition is seen to be illusory, and the hen 
is left helpless on the shore. Intuition, in fact, is an 
aspect and development of instinct, and, like all 
instinct, is admirable in those customary surroundings 
which have moulded the habits of the animal in 
question, but totally incompetent as soon as the 
surroundings are changed in a way which demands 
some non-habitual mode of action. 

The theoretical understanding of the world, which 
is the aim of philosophy, is not a matter of great 
practical importance to animals, or to savages, or 
even to most civilized men. It is hardly to be sup- 
posed, therefore, that the rapid, rough and ready 
methods of instinct or intuition will find in this field 
a favourable ground for their application. It is the 
older kinds of activity, which bring out our kinship 
with remote generations of animal and semi-human 
ancestors, that show intuition at its best. In such 
matters as self-preservation and love, intuition will 
act sometimes (though not always) with a swiftness 
and precision which are astonishing to the critical 
intellect. But philosophy is not one of the pursuits 
which illustrate our affinity with the past : it is a 
highly refined, highly civilized pursuit, demanding, 
for its success, a certain liberation from the life of 
instinct, and even, at times, a certain aloofness from 
all mundane hopes and fears. It is not in philosophy, 


therefore, that we can hope to see intuition at its 
best. On the contrary, since the true objects of 
philosophy, and the habits of thought demanded for 
their apprehension, are strange, unusual, and remote, 
it is here, more almost than anywhere else, that in- 
tellect proves superior to intuition, and that quick 
unanalysed convictions are least deserving of uncritical 

Before embarking upon the somewhat difficult and 
abstract discussions which lie before us, it will be well 
to take a survey of the hopes we may retain and the 
hopes we must abandon. The hope of satisfaction 
to our more human desires the hope of demonstrating 
that the world has this or that desirable ethical charac- 
teristic is not one which, so far as I can see, philosophy 
can do anything whatever to satisfy. The difference 
between a good world and a bad one is a difference 
in the particular characteristics of the particular 
things that exist in these worlds : it is not a sufficiently 
abstract difference to come within the province of 
philosophy. Love and hate, for example, are ethical 
opposites, but to philosophy they are dosely analogous 
attitudes towards objects. The general form and 
structure of those attitudes towards objects which 
constitute mental phenomena is a problem for philo- 
sophy ; but the difference between love and hate is not 
a difference of form or structure, and therefore belongs 
rather to the special science of psychology than to 
philosophy. Thus the ethical interests which have 
often inspired philosophers must remain in the back- 
ground : some kind of ethical interest may inspire 
the whole study, but none must obtrude in the detail 
or be expected in the special results which are sought. 

If this view seems at first sight disappointing, we 
may remind ourselves that a similar change has been 


found necessary in all the other sciences. The physi- 
cist or chemist is not now required to prove the ethical 
importance of his ions or atoms ; the biologist is not 
expected to prove the utility of the plants or animals 
which he dissects. In pre-scientific ages this was not 
the case. Astronomy, for example, was studied 
because men believed in astrology : it was thought 
that the movements of the planets had the most direct 
and important bearing upon the lives of human beings. 
Presumably, when this belief decayed and the dis- 
interested study of astronomy began, many who had 
found astrology absorbingly interesting decided that 
astronomy had too little human interest to be worthy 
of study. Physics, as it appears in Plato's Timaus 
for example, is full of ethical notions : it is an essential 
part of its purpose to show that the earth is worthy 
of admiration. The modern physicist, on the con- 
trary, though he has no wish to deny that the earth 
is admirable, is not concerned, as physicist, with its 
ethical tributes : he is merely concerned to find out 
facts, not to consider whether they are good or bad. 
In psychology, the scientific attitude is even more 
recent and more difficult than in the physical sciences : 
it is natural to consider that human nature is either 
good or bad, and to suppose that the difference between 
good and bad, so all-important in practice, must be 
important in theory also. It is only during the last 
century that an ethically neutral science of psychology 
has grown up; and here too ethical neutrality has 
been essential to scientific success. 

In philosophy, hitherto, ethical neutrality has been 
seldom sought and hardly ever achieved. Men have 
remembered their wishes, and have judged philosophies 
in relation to their wishes. Driven from the par- 
ticular sciences, the belief that the notions of good 


and evil must afford a key to the understanding of 
the world has sought a refuge in philosophy. But 
even from this last refuge, if philosophy is not to 
remain a set of pleasing dreams, this belief must be 
driven forth. It is a commonplace that happiness 
is not best achieved by those who seek it directly ; 
and it would seem that the same is true of the good. 
In thought, at any rate, those who forget good and 
evil and seek only to know the facts are more likely 
to achieve good than those who view the world through 
the distorting medium of their own desires. 

The immense extension of our knowledge of facts 
in recent times has had, as it had in the Renaissance, 
two effects upon the general intellectual outlook. 
On the one hand, it has made men distrustful of the 
truth of wide, ambitious systems : theories come and 
go swiftly, each serving, for a moment, to classify 
known facts and promote the search for new ones, 
but each in turn proving inadequate to deal with the 
new facts when they have been found. Even those 
who invent the theories do not, in science, regard them 
as anything but a temporary makeshift. The ideal 
of an all-embracing synthesis, such as the Middle 
Ages believed themselves to have attained, recedes 
further and further beyond the limits of what seems 
feasible. In such a world, as in the world of Mon- 
taigne, nothing seems worth while except the dis- 
covery of more and more facts, each in turn the death- 
blow to some cherished theory ; the ordering intellect 
grows weary, and becomes slovenly through despair. 

On the other hand, the new facts have brought new 
powers; man's physical control over natural forces 
has been increasing with unexampled rapidity, and 
promises to increase in the future beyond all easily 
assignable limits- ^ m alongside of despair ag 


regards ultimate theory there is an immense optimism 
as regards practice : what man can do seems almost 
boundless. The old fixed limits of human power, 
such as death, or the dependence of the race on an 
equilibrium of cosmic forces, are forgotten, and no 
hard facts are allowed to break in upon the dream 
of omnipotence. No philosophy is tolerated which 
sets bounds to man's capacity of gratifying his wishes ; 
and thus the very despair of theory is invoked to 
silence every whisper of doubt as regards the possi- 
bilities of practical achievement. 

In the welcoming of new fact, and in the suspicion 
of dogmatism as regards the universe at large, the 
modern spirit should, I think, be accepted as wholly 
an advance. But both in its practical pretensions 
and in its theoretical despair it seems to me to go 
too far. Most of what is greatest in man is called 
forth in response to the thwarting of his hopes by 
immutable natural obstacles; by the pretence of 
omnipotence, he becomes trivial and a little absurd. 
And on the theoretical side, ultimate metaphysical 
truth, though less all-embracing and harder of attain- 
ment than it appeared to some philosophers in the 
past, can, I believe, be discovered by those who are 
willing to combine the hopefulness, patience, and 
open-mindedness of science with something of the 
Greek feeling for beauty in the abstract world of 
logic and for the ultimate intrinsic value in the con- 
templation of truth. 

The philosophy, therefore, which is to be genuinely 
inspired by the scientific spirit, must deal with some- 
what dry and abstract matters, and must not hope 
to find an answer to the practical problems of life. To 
those who wish to understand much of what has in 
the past been most difficult and obscure in the constitu- 


tion of the universe, it has great rewards to offer 
triumphs as noteworthy as those of Newton and 
Darwin, and as important, in the long run, for the 
moulding of our mental habits. And it brings with 
it as a new and powerful method of investigation 
always does a sense of power and a hope of progress 
more reliable and better grounded than any that 
rests on hasty and fallacious generalization as to the 
nature of the universe at large. Many hopes which 
inspired philosophers in the past it cannot da-im to 
fulfil ; but other hopes, more purely intellectual, it 
can satisfy more fully than former ages could have 
deemed possible for human minds. 



THE topics we discussed in our first lecture, and the 
topics we shall discuss later, all reduce themselves, 
in so far as they are genuinely philosophical, to prob- 
lems of logic. This is not due to any accident, but 
to the fact that every philosophical problem, when it 
is subjected to the necessary analysis and purification, 
is found either to be not really philosophical at all, 
or else to be, in the sense in which we are using the 
word, logical. But as the word "logic" is never 
used in the same sense by two different philosophers, 
some explanation of what I mean by the word is 
indispensable at the outset. 

Logic, in the Middle Ages, and down to the present 
day in teaching, meant no more than a scholastic 
collection of technical terms and rules of syllogistic 
inference. Aristotle had spoken, and it was the part 
of humbler men merely to repeat the lesson after him. 
The trivial nonsense embodied in this tradition is still 
set in examinations, and defended by eminent authori- 
ties as an excellent " propaedeutic," i.e. a training in 
those habits of solemn humbug which are so great a 
help in later life. But it is not this that I mean to 
praise in saying that all philosophy is logic. Ever 
since the beginning of the seventeenth centuiy, all 


vigorous minds that have concerned themselves with 
inference have abandoned the mediaeval tradition, and 
in one way or other have widened the scope of logic. 

The first extension was the introduction of the 
inductive method by Bacon and Galileo by the 
former in a theoretical and largely mistaken form, 
by the latter in actual use in establishing the founda- 
tions of modern physics and astronomy. This is 
probably the only extension of the old logic which has 
become familiar to the general educated public. But 
induction, important as it is when regarded as a method 
of investigation, does not seem to remain when its 
work is done : in the final form of a perfected science, 
it would seem that everything ought to be deductive. 
If induction remains at all, which is a difficult question, 
it will remain merely as one of the principles according 
to which deductions are effected. Thus the ultimate 
result of the introduction of the inductive method 
seems not the creation of a new kind of non-deductive 
reasoning, but rather the widening of the scope of de- 
duction by pointing out a way of deducing which is 
certainly not syllogistic, and does not fit into the 
mediaeval scheme. 

The question of the scope and validity of induction 
is of great difficulty, and of great importance to our 
knowledge. Take such a question as, " Will the sun 
rise to-morrow ? " Our first instinctive feeling is 
that we have abundant reason for saying that it will, 
because it has risen on so many previous mornings. 
Now, I do not myself know whether this does afford 
a ground or not, but I am willing to suppose that it 
does. The question which then arises is : " What is 
the principle of inference by which we pass from past 
sunrises to future ones ? The answer given by Mill 
is that the inference depends upon the law of causation. 


Let us suppose this to be true ; then what is the 
reason for believing in the law of causation ? There 
are broadly three possible answers : (i) that it is itself 
known a priori ; (2) that it is a postulate ; (3) that 
it is an empirical generalization from past instances 
in which it has been found to hold. The theory that 
causation is known a priori cannot be definitely refuted, 
but it can be rendered very implausible by the mere 
process of formulating the law exactly, and thereby 
showing that it is immensely more complicated and 
less obvious than is generally supposed. The theory 
that causation is a postulate, i.e. that it is something 
which we choose to assert although we know that it 
is very likely false, is also incapable of refutation ; but 
it is plainly also incapable of justifying any use of the 
law in inference. We are thus brought to the theory 
that the law is an empirical generalization, which is 
the view held by Mill. 

But if so, how are empirical generalizations to be 
justified ? The evidence in their favour cannot be 
empirical, since we wish to argue from what has been 
observed to what has not been observed, which can only 
be done by means of some known relation of the 
observed and the unobserved; but the unobserved, 
by definition, is not known empirically, and therefore 
its relation to the observed, if known at all, must be 
known independently of empirical evidence. Let us 
see what Mill says on this subject. 

According to Mill, the law of causation is proved by 
an admittedly fallible process called "induction by 
simple enumeration." This process, he says, " con- 
sists in ascribing the nature of general truths to all 
propositions which are true in every instance that we 
happen to know of." J As regards its fallibility, he 
* Logic, Book III. chapter ill. 2. 


asserts that r< the precariousness of the method of 
simple enumeration is in an inverse ratio to the large- 
ness of the generalization. The process is delusive and 
insufficient, exactly in proportion as the subject-matter 
of the observation is special and limited in extent. 
As the sphere widens, this unscientific method becomes 
less and less liable to mislead ; and the most universal 
class of truths, the law of causation for instance, and 
the principles of number and of geometry, are duly and 
satisfactorily proved by that method alone, nor are 
they susceptible of any other proof," x 

In the above statement, there are two obvious 
lacunae : (i) How is the method of simple enumeration 
itself justified? (2) What logical principle, if any, 
covers the same ground as this method, without 
being liable to its failures ? Let us take the second 
question first. 

A method of proof which, when used as directed, 
gives sometimes truth and sometimes falsehood as 
the method of simple enumeration does is obviously 
not a valid method, for validity demands invariable 
truth. Thus, if simple enumeration is to be rendered 
valid, it must not be stated as Mill states it. We shall 
have to say, at most, that the data render the result 
probable. Causation holds, we shall say, in every 
instance we have been able to test ; therefore it probably 
holds in untested instances. There are terrible diffi- 
culties in the notion of probability, but we may ignore 
them at present, We thus have what at least may 
be a logical principte, since it is without exception. 
If a proposition is true in every instance that we happen 
to know of, and if the instances are very numerous, 
then, we shall say, it becomes very probable, on the 
data, that it will be true in any further instance. This 
x Book III. chapter acri. 3. 


is not refuted by the fact that what we declare to be 
probable does not always happen, for an event may be 
probable on the data and yet not occur. It is, however, 
obviously capable of further analysis, and of more 
exact statement. We shall have to say something 
like this : that every instance of a proposition z being 
true increases the probability of its being true in a fresh 
instance, and that a sufficient number of favourable 
instances will, in the absence of instances to the contrary, 
make the probability of the truth of a fresh instance 
approach indefinitely near to certainty. Some such 
principle as this is required if the method of simple 
enumeration is to be valid. 

But this brings us to our other question, namely, 
how is our principle known to be true ? Obviously, 
since it is required to justify induction, it cannot be 
proved by induction ; since it goes beyond the empirical 
data, it cannot be proved by them alone ; since it is 
required to justify all inferences from empirical data 
to what goes beyond them, it cannot itself be even 
rendered in any degree probable by such data. Hence, 
if it is known, it is not known by experience, but 
independently of experience. I do not say that any 
such principle is known : I only say that it is required 
to justify the inferences from experience which empiri- 
cists allow, and that it cannot itself be justified 
empirically. 3 

A similar conclusion can be proved by similar 
arguments concerning any other logical principle. 
Thus logical knowledge is not derivable from experi- 
ence alone, and the empiricist's philosophy can 
therefore not be accepted in its entirety, in spite 

K Or rather a prepositional function. 
* The subject of causality and induction will be discussed 
again in Lecture VIII. 


of its excellence in many matters which He outside 

Hegel and his followers widened the scope of logic 
in quite a different way a way which I believe to be 
fallacious, but which requires discussion if only to show 
how their conception of logic differs from the con- 
ception which I wish to advocate. In their writings, 
logic is practically identical with metaphysics. In 
broad outline, the way this came about is as follows. 
Hegel believed that, by means of a priori reasoning, 
it could be shown that the world must have various 
important and interesting characteristics, since any 
world without these characteristics would be impossible 
and self-contradictory. Thus what he calls "logic" 
is an investigation of the nature of the universe, in so 
far as this can be inferred merely from the principle 
that the universe must be logically self-consistent. 
I do not myself believe that from this principle alone 
anything of importance can be inferred as regards the 
existing universe. But, however that may be, I 
should not regard Hegel's reasoning, even if it were 
valid, as properly belonging to logic : it would rather 
be an application of logic to the actual world. Logic 
itself would be concerned rather with such questions 
as what self-consistency is, which Hegel, so far as I 
know, does not discuss. .And though he criticizes the 
traditional logic, and professes to replace it by an 
improved logic of his own, there is some sense in which 
the traditional logic, with all its faults, is uncritically 
and unconsciously assumed throughout his reasoning. 
It is not in the direction advocated by him, it seems to 
me, that the reform of logic is to be sought, but^by a 
more fundamental, more patient, and less ambitious 
investigation into the presuppositions which his system 
shares with those of most other philosophers. 


The way in which, as it seems to me, Hegel's system 
assumes the ordinary logic which it subsequently 
criticizes, is exemplified by the general conception of 
"categories" with which he operates throughout. 
This conception is, I think, essentially a product of 
logical confusion, but it seems in some way to stand 
for the conception of " qualities of Reality as a whole." 
Mr. Bradley has worked out a theory according to which, 
in all judgment, we are ascribing a predicate to Reality 
as a whole ; and this theory is derived from Hegel. 
Now the traditional logic holds that every proposition 
ascribes a predicate to a subject, and from this it easily 
follows that there can be only one subject, the Absolute, 
for if there were two, the proposition that there were two 
would not ascribe a predicate to either. Thus Hegel's 
doctrine, that philosophical propositions must be of 
the form, " the Absolute is such-and-such," depends 
upon the traditional belief in the universality of the 
subject-predicate fonn. This belief, being traditional, 
scarcely self-conscious, and not supposed to be impor- 
tant, operates underground, and is assumed in argu- 
ments which, like the refutation of relations, appear 
at first sight such as to establish its truth. This is 
the most important respect in which Hegel uncritically 
assumes the traditional logic. Other less important 
respects though important enough to be the source 
of such essentially Hegelian conceptions as the " con- 
crete universal " and the " union of identity in differ- 
ence "will be found where he explicitly deals with 
formal logic. 1 

* See the translation by EL S. Macran, Hegel's Doctrine of 
Formal Logic, Oxford, 1912. Hegel's argument in this 
portion of bis " Logic " depends throughout upon confusing 
the " is " of predication, as in " Socrates is mortal," with the 
" is " of identity, as in " Socrates is the philosopher who drank 


There is quite another direction in which a large 
technical development of logic has taken place : I 
mean the dkection of what is called logistic or mathe- 
matical logic This kind of logic is mathematical in 
two different senses : it is itself a branch of mathe- 
matics, and it is the logic which is specially applicable 
to other more traditional branches of mathematics, 
Historically, it began as merely a branch of mathematics: 
its special applicability to other branches is a more 
recent development. In both respects, it is the fulfil- 
ment of a hope which Leibniz cherished throughout his 
life, and pursued with all the ardour of his amazing 
intellectual energy. Much of his work on this subject 
has been published recently, since his discoveries have 
been remade by others ; but none was published by 
him, because his results persisted in contradicting 
certain points in the traditional doctrine of the 
syllogism. We now know that on these points the 
traditional doctrine is wrong, but respect for Aristotle 
prevented Leibniz from realizing that this was possible. 1 

The modern development of mathematical logic 

the hemlock. 1 ' Owing to tbfa confusion, he ^Tiinlrg that 
" Socrates " and " mortal" must be identical. Seeing that 
they are different, he does not infer, as others would, that there 
is a mistake somewhere, but that they exhibit " identity in 
difference." Again, Socrates is particular, "mortal" is 
universal. Therefore, he says, since Socrates is mortal, it 
follows that the particular is the universal taking the " is " 
to be throughout expressive of identity. But to say " the 
particular is the universal" is self -contradictory. Again 
Hegel does not suspect a mistake but proceeds to synthesize 
particular and universal in the individual, or concrete universal. 
This is an example of how, for want of care at the start, vast 
and imposing systems of philosophy axe built upon stupid and 
trivial confusions, which, but for the almost incredible fact 
that they are unintentional, one would be tempted to charac-. 
terize as puns. 

* Cf. Couturat, La Logique de Leibniz, pp. 361, 386. 



dates from Boole's Laws of Thought (1854). But in 
him and his successors, before Peano and Frege, the 
only thing really achieved, apart from certain details, 
was the invention of a mathematical symbolism for 
deducing consequences from the premisses which the 
newer methods shared with those of Aristotle. This 
subject has considerable interest as an independent 
branch of mathematics, but it has very little to do with 
real logic. The first serious advance in real logic since 
the time of the Greeks was made independently by 
Peano and Frege both mathematicians. They both 
arrived at their logical results by an analysis of mathe- 
matics. Traditional logic regarded the two propositions, 
" Socrates is mortal " and " All men are mortal," as 
being of the same form ; * Peano and Frege showed 
that they are utterly different in form. The philosophical 
importance of logic may be illustrated by the fact that 
this confusion which is still committed by most 
writers obscured not only the whole study of the 
forms of judgment and inference, but also the relations 
of things to their qualities, of concrete existence to 
abstract concepts, and of the world of sense to the world 
of Platonic ideas. Peano and Frege, who pointed out 
the error, did so for technical reasons, and applied their 
logic mainly to technical developments ; but the 
philosophical importance of the advance which they 
made is impossible to exaggerate. 

Mathematical logic, even in its most modern form, 
is not directly of philosophical importance except 
in its beginnings. After the beginnings, it belongs 
rather to mathematics than to philosophy. Of its 
beginnings, which are the only part of it that can 

* It was often recognized that there was some difference 
between them, but it was not recognized that the difference 
is fundamental, and of very great importance. 


properly be called philosophical logic, I shall speak 
shortly. But even the later developments, though 
not directly philosophical, will be f ound of great indirect 
use in philosophizing. They enable us to deal easily 
with more abstract conceptions than merely verbal 
reasoning can enumerate ; they suggest fruitful hypo- 
theses which otherwise could hardly be thought of ; 
and they enable us to see quickly what is the smallest 
store of materials with which a given logical or scientific 
edifice can be constructed. Not only Frege's theory 
of number, which we shall deal with in Lecture VII, 
but the whole theory of physical concepts which will 
be outlined in our next two lectures, is inspired by 
mathematical logic, and could never have been 
imagined without it. 

In both these cases, and in many others, we shall 
appeal to a certain principle called " the principle of 
abstraction." This principle, which might equally well 
be called " the principle which dispenses with abstrac- 
tion," and is one which clears away incredible accumu- 
lations of metaphysical lumber, was directly suggested 
by mathematical logic, and could hardly have been 
proved or practically used without its help. The 
principle will be explained in our fourth lecture, but 
its use may be briefly indicated in advance. When a 
group of objects have that kind of similarity which 
we are inclined to attribute to possession of a common 
quality, the principle in question shows that membership 
of the group will serve all the purposes of the supposed 
common quality, and that therefore, unless some 
common quality is actually known, the group or class 
of gin-nlar objects may be used to replace the common 
quality, which need not be assumed to exist. In this 
and other ways, the indirect uses of even the later parts 
of mathematical logic are very great ; but it is now 


time to turn our attention to its philosophical founda- 

In every proposition and in every inference there 
is, besides the particular subject-matter concerned, 
a certain form, a way in which the constituents of the 
proposition or inference are put together. If I say, 
" Socrates is mortal," " Jones is angry," " The sun is 
hot," there is something in common in these three 
cases, something indicated by the word " is." What 
is in common is the form of the proposition, not an 
actual constituent. If I say a number of things about 
Socrates that he was an Athenian, that he married 
Xantippe, that he drank the hemlock there is a 
common constituent, namely Socrates, in all the propo- 
sitions I enunciate, but they have diverse forms. If, 
on the other hand, I take any one of these propositions 
and replace its constituents, one at a time, by other 
constituents, the form remains constant, but no con- 
stituent remains. Take (say) the series of propositions, 
"Socrates drank the hemlock," "Coleridge drank 
the hemlock/' " Coleridge drank opium," " Coleridge 
ate opium." The form remains unchanged throughout 
this series, but all the constituents are altered. Thus 
form is not another constituent, but is the way the 
constituents axe put together. It is forms, in this 
sense, that are the proper object of philosophical 

It is obvious that the knowledge of logical forms 
is something quite different from knowledge of existing 
things. The form of " Socrates drank the hemlock " 
is not an existing thing like Socrates or the hemlock, 
nor does it even have that dose relation to existing 
things that drinking has. It is something altogether 
more abstract and remote. We might understand all 
the separate words of a sentence without understanding 


the sentence : if a sentence is long and complicated, 
this is apt to happen. In such a case we have knowledge 
of the constituents, but not of the form. We may also 
have knowledge of the f orm without having knowledge 
of the constituents. If I say, " Rorarius drank the 
hemlock/ 1 those among you who have never heard of 
Rorarius (supposing there are any) will understand the 
form, without having knowledge of all the constituents. 
In order to understand a sentence, it is necessary to 
have knowledge berth of the constituents and of the 
particular instance of the form. It is in this way that 
a sentence conveys iaf onnation, since it tells us that 
certain known objects are related according to a certain 
known form. Thus some kind of knowledge of logical 
forms, though with most people it is not explicit, is 
involved in all understanding of discourse. It is the 
business of philosophical logic to extract this knowledge 
from its concrete integuments, and to render it explicit 
and pure. 

In all inference, form alone is essential : the particu- 
lar subject-matter is irrelevant except as securing the 
truth of the premisses. This is one reason for the 
great importance of logical form. When I say, 
" Socrates was a man, all men are mortal, therefore 
Socrates was mortal," the connection of premisses 
and conclusion does not in any way depend upon its 
being Socrates and man and mortality that I am 
mentioning. The general form of the inference may be 
expressed in some such words as : " If a thing has a 
certain property, and whatever has this property has 
a certain other property, then the thing in question 
also has that other property." Here no particular 
things or properties axe mentioned : the proposition 
is absolutely general. All inferences, when stated 
fully, are instances of propositions having this kind of 


generality. If they seem to depend upon the subject- 
matter otherwise than as regards the truth of the 
premisses, that is because the premisses have not been 
all explicitly stated. In logic, it is a waste of time to 
deal with inferences concerning particular cases : 
we deal throughout with completely general and purely 
formal implications, leaving it to other sciences to 
discover when the hypotheses axe verified and when 
they are not. 

But the forms of propositions giving rise to inferences 
are not the simplest forms ; they are always hypo- 
thetical, stating that if one proposition is true, then 
so is another. Before considering inference, there- 
fore, logic must consider those simpler forms which 
inference presiipposes. Here the traditional logic 
failed completely: it believed that there was only 
one form of simple proposition (i.e. of proposition 
not stating a relation between two or more other 
propositions), namely, the form which ascribes a 
predicate to a subject. This is the appropriate form 
in assigning the qualities of a given thing we may 
say " this thing is round, and red, and so on." Gram- 
mar favours this form, but philosophically it is so far 
from universal that it is not even very common. If 
we say " this thing is bigger than that," we are not 
assigning a mere quality of " this," but a relation of 
"this" and "that." We might express the same 
fact by saying " that thing is smaller than this," where 
grammatically the subject is changed. Thus propo- 
sitions stating that two things have a certain relation 
have a different form from subject-predicate propo- 
sitions, and the failure to perceive this difference or 
to allow for it has been the source of many errors in 
traditional metaphysics. 

The belief or unconscious conviction that all propo 


sitions are of the subject-predicate form in other 
words : that every fact consists in some thing having 
some quality has rendered most philosophers incapable 
of giving any account of the world of science and daily 
life. If they had been honestly anxious to give such 
an account, they would probably have discovered 
their error very quickly ; but most of them were less 
anxious to understand the world of science and daily 
life, than to convict it of unreality in the interests 
of a super-sensible "real" world. Belief in the 
unreality of the world of sense arises with irresistible 
force in certain moods moods which, I imagine, have 
some simple physiological basis, but are none the 
less powerfully persuasive. The conviction born of 
these moods is the source of most mysticism and 
of most metaphysics. When the emotional intensity of 
such a mood subsides, a man who is in the habit of 
reasoning will search for logical reasons in favour 
of the belief which he finds in himself. But since the 
belief already exists, he will be very hospitable to any 
reason that suggests itself. The paradoxes apparently 
proved by his logic are really the paradoxes of mysticism, 
and are the goal which he feds his logic must reach 
if it is to be in accordance with insight. It is in this 
way that logic has been pursued by those of the great 
philosophers who were mystics notably Plato, Spinoza, 
and Hegel. But since they usually took for granted 
the supposed insight of the mystic emotion, their 
logical doctrines were presented with a certain dryness, 
and were believed by their disciples to be quite inde- 
pendent of the sudden jl.nnTnTjna.fifm from which they 
sprang. Nevertheless their origin dung to them, and 
they remained to borrow a useful word from Mr. 
Santayana " malicious " in regard to the world of 
science and common sense. It is only so that we 


can account for the complacency with which philo- 
sophers have accepted the inconsistence of their 
doctrines with all the common and scientific facts 
which seem best established and most worthy of belief. 

The logic of mysticism shows, as is natural, the 
defects which are inherent in anything malicious. 
While the mystic mood is dominant, the need of logic 
is not felt ; as the mood fades, the impulse to logic 
reasserts itself, but with a desire to retain the vanishing 
insight, or at least to prove that it was insight, and 
that what seems to contradict it is illusion. The logic 
which thus arises is not quite disinterested or candid, 
and is inspired by a certain hatred of the daily world 
to which it is to be applied. Such an attitude naturally 
does not tend to the best results. Everyone knows 
that to read an author simply in order to refute him 
is not the way to understand him ; and to read the 
book of Nature with a conviction that it is all illusion 
is just as unlikely to lead to understanding. If our 
logic is to find the common world intelligible, it must 
not be hostile, but must be inspired by a genuine 
acceptance such as is not usually to be found among 

Traditional logic, since it holds that all propositions 
have the subject-predicate form, is unable to admit 
the reality of relations : all relations, it maintains, 
must be reduced to properties of the apparently related 
terms. There are many ways of refuting this opinion ; 
one of the easiest is derived from the consideration 
of what are called " asymmetrical " relations. In 
order to explain this, I will first explain two independent 
ways of classifying relations 

Some relations, when they hold between A and B, 
also hold between B and A. Such, for example, is 
the relation " brother or sister." If A is a brother or 


sister of B thsn B is a brother or sister of A. Such 
again is any kind of similarity, say similarity of colour. 
Any kind of dissimilarity is also of this kind : if the 
colour of A is unlike the colour of B, then the colour of 
B is unlike the colour of A. Relations of this sort are 
called symmetrical. Thus a relation is symmetrical 
if, whenever it holds between A and B, it also holds 
between B and A. 

All relations that are not symmetrical are called 
non-symmetrical. Thus " brother " is ncni-syimnetrical, 
because, if A is a brother of B, it may happen that 
B is a sister of A. 

A relation is called asymmetrical when, if it holds 
between A and B, it never holds between B and A. 
Thus husband, father, grandfather, etc., are asym- 
metrical relations. So are before, after, greater, above, 
to the right of, etc. All the relations that give rise to 
series are of this kind. 

Classification into symmetrical, asymmetrical and 
merely non-symmetrical relations is the first of the 
two classifications we had to consider. The second 
is into transitive, intransitive, and merely non-transitive 
relations, which are defined as follows. 

A relation is said to be transitive, if, whenever it 
holds between A and B and also between B and C, 
it holds between A and C. Thus before, after, greater, 
above are transitive, All relations giving rise to series 
are transitive, but so are many others. The transitive 
relations just mentioned were asymmetrical, but 
many transitive relations are symmetrical f or instance, 
equality in any respect, exact identity of colour, being 
equally numerous (as applied to collections), and 
so on. 

A relation is said to be non-transitive whenever it 
is not transitive. Thus " brother " is non-transitive, 


because a brother of one's brother may be oneself. 
All kinds of dissimilarity are non-transitive. 

A relation is said to be intransitive when, if A has the 
relation to B, and B to C, A never has it to C. Thus 
" father " is intransitive. So is such a relation as 
" one inch taller " or " one year later." 

Let us now, in the light of this classification, return 
to the question whether all relations can be reduced 
to predications. 

In the case of symmetrical relations i.e. relations 
which, if they hold between A and B, also hold between 
B and A some kind of plausibility can be given to 
this doctrine. A symmetrical relation which is 
transitive, such as equality, can be regarded as expres- 
sing possession of some common property, while one 
which is not transitive, such as inequality, can be 
regarded as expressing possession of different properties. 
But when we come to asymmetrical relations, such as 
before and after, greater and less, etc., the attempt 
to reduce them to properties becomes obviously 
impossible. When, for example, two things are merely 
known to be unequal, without our knowing which 
is greater, we may say that the inequality results 
from their having different magnitudes, because 
inequality is a symmetrical relation ; but to say that 
when one thing is greater than another, and not merely 
unequal to it, that means that they have different 
magnitudes, is formally incapable of explaining the 
facts. For if the other thing had been greater than 
the one, the magnitudes would also have been different, 
though the fact to be explained would not have been 
the same. Thus mere difference of magnitude is not all 
that is involved, since, if it were, there would be no 
difference between one thing being greater than another, 
and the other being greater than the one. We shall 


have to say that the one magnitude is greater than 
the other , and thus we shall have failed to get rid of 
the relation " greater." In short, both possession of 
the same property and possession of different properties 
are symmetrical relations, and therefore cannot account 
for the existence of asymmetrical relations. 

Asymmetrical relations are involved in all series 
in space and time, greater and less, whole and part, 
and many others of the most important characteristics 
of the actual world. All these aspects, therefore, the 
logic which reduces everything to subjects and predi- 
cates is compelled to condemn as error and mere 
appearance. To those whose logic is not malicious, 
such a wholesale condemnation appears impossible. 
And in fact there is no reason except prejudice, so far 
as I can discover, for denying the reality of relations. 
When once their reality is admitted, all kgical grounds 
for supposing the world of sense to be illusory disappear. 
If this is to be supposed, it must be frankly and simply 
on the ground of mystic insight unsupported by 
argument. It is impossible to argue against what 
professes to be insight, so long as it does not argue in 
its own favour. As logicians, therefore, we may 
admit the possibility of the mystic's world, while yet, 
so long as we do not have his insight, we must continue 
to study the everyday world with which we are 
familiar. But when he contends that our world is 
impossible, then our logic is ready to repel his attack. 
And the first step in creating the logic which is to 
perform this service is the recognition of the reality of 

Relations which have two terms are only one kind 
of relations. A relation may have three terms, or four, 
or any number. Relations of two terms, being the 
simplest, have received more attention than the 


others, and have generally been alone considered by 
philosophers, both those who accepted and those 
who denied the reality of relations. But other relations 
have their importance, and are indispensable in the 
solution of certain problems. Jealousy, for example, 
is a relation between three people. Professor Royce 
mentions the relation " giving " : when A gives B 
to C, that is a relation of three terms. 1 When a man 
says to his wife : " My dear, I wish you could induce 
Angelina to accept Edwin," his wish constitutes a 
relation between four people, himself, his wife, Angelina, 
and Edwin. Thus such relations are by no means 
recondite or rare. But in order to explain exactly 
how they differ from relations of two terms, we must 
embark upon a classification of the logical forms of 
facts, which is the first business of logic, and the 
business in which the traditional logic has been most 

The existing world consists of many things with 
many qualities and relations. A complete description 
of the existing world would require not only a catalogue 
of the things, but also a mention of all their qualities 
and relations. We should have to know not only this 
that, and the other thing, but also which was red, 
which yellow, which was earlier than which, which was 
which between two others, and so on. When I speak 
of a " fact," I do not mean one of the simple things 
in the world ; I mean that a certain thing has a certain 
quality, or that certain things have a certain relation. 
Thus, for example, I should not call Napoleon a fact, 
but I should call it a fact that he was ambitious, or 
that he married Josephine. Now a fact, in this sense, 
is never simple, but always has two or more constitu- 
ents. When it simply assigns a quality to a thing, 

Encyclopedia of the Philosophical Sciences, vol. i. p. 97. 


it has only two constituents, the thing and the quality. 
When it consists of a relation between two things, 
it has three constituents, the things and the relation. 
When it consists of a relation between three things, 
it has four constituents, and so on. The constituents 
of facts, in the sense in which we are using the 
word " fact," are not other facts, but are things 
and qualities or relations. When we say that there 
are relations of more than two terms, we mean that 
there are single facts consisting of a single relation 
and more than two things. I do not mean that 
one relation of two terms may hold between A and 
B, and also between A and C, as, for example, a 
man is the son of his father and also the son of his 
mother. This constitutes two distinct facts: if we 
choose to treat it as one fact, it is a fact which has 
facts for its constituents. But the facts I am speaking 
of have no facts among their constituents, but only 
things and relations. For example, when A is jealous 
of B on account of C, there is only one fact, involving 
three people ; there are not two instances of jealousy, 
but only one. It is in such cases that I speak of a 
relation of three terms, where the simplest possible 
fact in which the relation occurs is one involving three 
things in addition to the relation. And the same 
applies to relations of four terms or five or any other 
number. All such relations must be admitted in 
our inventory of the logical forms of facts : two facts 
involving the same number of things have the same 
form, and two which involve different numbers of 
things have different forms. 

Given any fact, there is an assertion which expresses 
the fact. The fact itself is objective, and independent 
of our thought or opinion about it ; but the assertion 
is something which involves thought, and may be 


either true or false. An assertion may be positive or 
negative : we may assert that Charles I was executed, 
or that he did not die in his bed. A negative assertion 
may be said to be a denial. Given a form of words 
which must be either true or false, such as " Charles I 
died in his bed," we may either assert or deny this 
form of words : in the one case we have a positive 
assertion, in the other a negative one. A form of 
words which must be either true or false I shall call 
a proposition. Thus a proposition is the same as what 
may be significantly asserted or denied. A proposition 
which expresses what we have called a fact, i.e. which, 
when asserted, asserts that a certain thing has a 
certain quality, or that certain things have a certain 
relation, will be called an atomic proposition, because, 
as we shall see immediately, there are other propositions 
into which atomic propositions enter in a way analogous 
to that in which atoms enter into molecules. Atomic 
propositions, although, like facts, they may have any 
one of an infinite number of forms, are only one kind 
of propositions. All other kinds are more complicated. 
In order to preserve the parallelism in language 
as regards facts and propositions, we shall give the name 
" atomic facts " to the facts we have hitherto been 
considering. Thus atomic facts are what determine 
whether atomic propositions are to be asserted or 

Whether an atomic proposition, such as "this is 
red," or "this is before that," is to be asserted or 
denied can only be known empirically. Perhaps one 
atomic fact may sometimes be capable of being inferred 
from another, though this seems very doubtful ; but 
in any case it cannot be inferred from premisses no 
one of which is an atomic fact. It follows that, if 
atomic facts are to be known at all, some at least must 


be known without inference. The atomic facts which 
we come to know in this way are the facts of sense- 
perception; at any rate, the facts of sense-percep- 
tion are those which we most obviously and 
certainly come to know in this way. If we knew all 
atomic facts, and also knew that there were none 
except those we knew, we should, theoretically, be 
able to infer all truths of whatever form. 1 Thus logic 
would then supply us with the whole of the apparatus 
required. But in the first acquisition of knowledge 
concerning atomic facts, logic is useless. In pure logic, 
no atomic fact is ever mentioned : we confine ourselves 
whofly to forms, without asking ourselves what objects 
can fill the forms. Thus pure logic is independent of 
atomic facts ; but conversely, they are, in a sense, 
independent of logic. Pure logic and atomic facts 
are the two poles, the wholly a priori and the wholly 
empirical. But between the two lies a vast intermediate 
region, which we must now briefly explore. 

" Molecular " propositions are such as contain con- 
junctions if, or, and, unless, etc. and such words 
are the marks of a molecular proposition. Consider 
such an assertion as, " If it rains, I shall bring my 
umbrella." This assertion is just as capable of truth 
or falsehood as the assertion of an atomic proposition, 
but it is obvious that either the corresponding fact, 
or the nature of the correspondence with fact, must 
be quite different from what it is in the case of an atomic 
proposition. Whether it rains, and whether I bring 
my umbrella, are each severally matters of atomic 

* This perhaps requires modification in order to include 
such facts as beliefs and wishes, since such facts apparently 
contain propositions as components. Such facts, though not 
strictly atomic, must be supposed included if the statement 
in the text is to be true. 


fact, ascertainable by observation. But the connection 
of the two involved in saying that if the one happens, 
then the other will happen, is something radically 
different from either of the two separately. It does not 
require for its truth that it should actually rain, or that 
I should actually bring my umbrella; even if the 
weather is cloudless, it may still be true that I should 
have brought my umbrella if the weather had been 
different. Thus we have here a connection of two 
propositions, which does not depend upon whether 
they are to be asserted or denied, but only upon the 
second being inferable from the first. Such propositions, 
therefore, have a f oim which is different from that of 
any atomic proposition. 

Such propositions are important to logic, because all 
inference depends upon them. If I have told you that if 
it rains I shall bring my umbrella, and if you see that 
there is a steady downpour, you can infer that I shall 
bring my umbrella. There can be no inference except 
where propositions are connected in some such way, 
so that from the truth or falsehood of the one something 
follows as to the truth or falsehood of the other. It 
seems to be the case that we can sometimes know 
molecular propositions, as in the above instance of 
the umbrella, when we do not know whether the 
component atomic propositions are true or false. The 
practical utility of inference rests upon this fact. 

The next kind of propositions we have to consider 
are general propositions, such as " all men are mortal/' 
" all equilateral triangles are equiangular." And with 
these belong propositions in which the word " some " 
occurs, such as " some men are philosophers " or " some 
philosophers are not wise." These are the denials of 
general propositions, namely (in the above instances), 
of " all men are non-philosophers " and " all philoso- 


pliers are wise." We will call propositions containing the 
word" some " negative general propositions, and those 
containing the word " all " positive general propositions. 
These propositions, it will be seen, begin to have the 
appearance of the propositions in logical text-books. 
But their peculiarity and complexity are not known 
to the text-books, and the problems which they raise 
are only discussed in the most superficial ma.nTip.r- 

When we were discussing atomic facts, we saw that 
we should be able, theoretically, to infer all other 
truths by logic if we knew all atomic facts and also knew 
that there were no other atomic facts besides those we 
knew. The knowledge that there are no other atomic 
facts is positive general knowledge ; it is the knowledge 
that " all atomic facts are known to me," or at least 
" all atomic facts are in this collection "however the 
collection may be given. It is easy to see that general 
propositions, such as " all men are mortal," cannot be 
known by inference from atomic facts alone. If we 
could know each individual man, and know that he 
was mortal, that would not enable us to know that all 
men are mortal, unless we knew that those were all the 
men there are, which is a general proposition. If we 
knew every other existing thing throughout the universe, 
and knew that each separate thing was not an immortal 
man, that would not give us our result unless we knew 
that we had explored the whole universe, i.e. unless 
we knew " all things belong to this collection of things 
I have examined." Thus general truths cannot be 
inferred from particular truths alone, but must, if 
they are to be known, be either self-evident or inferred 
from premisses of which at least one is a general truth. 
But all empirical evidence is of particular truths. 
Hence, if there is any knowledge of general truths at 
all, there must be some knowledge of general truths 



which is independent of empirical evidence, i.e. does 
not depend upon the data of sense. 

The above conclusion, of which we had an instance 
in the case of the inductive principle, is important, 
since it affords a refutation of the older empiricists. 
They believed that all our knowledge is derived from 
the senses and dependent upon them. We see that, 
if this view is to be maintained, we must refuse to admit 
that we know any general propositions. It is perfectly 
possible logically that this should be the case, but it 
does not appear to be so in fact, and indeed no one 
would dream of maintaining such a view except a 
theorist at the last extremity. We must therefore 
admit that there is general knowledge not derived from 
sense, and that some of this knowledge is not obtained 
by inference but is primitive. 

Such general knowledge is to be found in logic. 
Whether there is any such knowledge not derived 
from logic, I do not know ; but in logic, at any rate, 
we have such knowledge. It will be remembered that 
we excluded from pure logic such propositions as, 
"Socrates is a man, all men are mortal, therefore 
Socrates is mortal," because Socrates and man and 
mortal are empirical terms, only to be understood 
through particular experience. The corresponding 
proposition in pure logic is : " If anything has a 
certain property, and whatever has this property 
has a certain other property, then the thing in 
question has the other property." This proposition is 
absolutely general: it applies to all things and all 
properties. And it is quite self-evident. Thus in such 
propositions of pure logic we have the self-evident 
general propositions of which we were in search. 

A proposition such as " If Socrates is a man, and all 
men are mortal, then Socrates is mortal," is true in 

virtue of its form alone. Its truth, in this hypothetical 
form, does not depend upon whether Socrates actually 
is a man, nor upon whether in fact all men are mortal ; 
thus it is equally true when we substitute other terms 
for Socrates and man and mortal. The general truth 
of which it is an instance is purely formal, and belongs 
to logic. Since this general truth does not mention 
any particular thing, or even any particular quality 
or relation, it is wholly independent of the accidental 
facts of the existent world, and can be known, theo- 
retically, without any experience of particular things 
or their qualities and relations. 

Logic, we may say, consists of two parts. The first 
part investigates what propositions are and what 
forms they may have; this part enumerates the 
different kinds of atomic propositions, of molecular 
propositions, of general propositions, and so on. The 
second part consists of certain supremely general 
propositions, which assert the truth of all propositions 
of certain forms. This second part merges into pure 
mathematics, whose propositions all turn out, on 
analysis, to be such general formal truths. The first 
part, which merely enumerates forms, is the more 
difficult, and philosophically the more important; 
and it is the recent progress in this first part, more 
than anything else, that has rendered a truly scientific 
discussion of many philosophical problems possible. 

The problem of the nature of judgment or belief 
may be taken as an example of a problem whose 
solution depends upon an adequate inventory of logical 
forms. We have already seen how the supposed 
universality of the subject-predicate form made it 
impossible to give a right analysis of serial order, and 
therefore made space and time unintelligible. But in 
this case it was only necessary to admit relations of 


two terms. The case of judgment demands the admis- 
sion of more complicated forms. If all judgments were 
true, we might suppose that a judgment consisted in 
apprehension of zfact, and that the apprehension was 
a relation of a mind to the fact. From poverty in the 
logical inventory, this view has often been held. But 
it leads to absolutely insoluble difficulties in the case 
of error. Suppose I believe that Charles I died in his 
bed. There is no objective fact " Charles I's death in 
his bed" to which I can have a relation of appre- 
hension. Charles I and death and his bed are objective, 
but they are not, except in my thought, put together 
as my false belief supposes. It is therefore necessary, 
in analysing a belief, to look for some other logical 
form than a two-term relation. Failure to realize this 
necessity has, in my opinion, vitiated almost everything 
that has hitherto been written on the theory of know- 
ledge, making the problem of error insoluble and the 
difference between belief and perception inexplicable. 

Modern logic, as I hope is now evident, has the effect 
of enlarging our abstract imagination, and providing 
an infinite number of possible hypotheses to be applied 
in the analysis of any complex fact. In this respect 
it is the exact opposite of the logic practised by the 
classical tradition. In that logic, hypotheses which 
seem prima fade possible are professedly proved 
impossible, and it is decreed in advance that reality 
must have a certain special character. In modern 
logic, on the contrary, while the prima fade hypotheses 
as a rule remain admissible, others, which only logic 
would have suggested, axe added to our stock, and are 
very often found to be indispensable if a right analysis 
of the facts is to be obtained. The old logic put thought 
in fetters, while the new logic gives it wings. It has, 
in my opinion, introduced the same kind of advance 


into philosophy as Galileo introduced into physics, 
making it possible at last to see what kinds of problems 
may be capable of solution, and what kinds must be 
abandoned as beyond human powers. And where a 
solution appears possible, the new logic provides a 
method which enables us to obtain, results that do not 
merely embody personal idiosyncrasies, but must 
command the assent of all who are competent to form 
an opinion. 



PHILOSOPHY may be approached by many roads, but 
one of .the oldest and most travelled is the road which 
leads through doubt as to the reality of the world of 
sense. In Indian mysticism, in Greek and modern 
monistic philosophy from Parmenides onward, in 
Berkeley, in modern physics, we find sensible appear- 
ance criticized and condemned for a bewildering 
variety of motives. The mystic condemns it on the 
ground of immediate knowledge of a more real and 
significant world behind the veil ; Parmenides and 
Plato condemn it because its continual flux is thought 
inconsistent with the unchanging nature of the abstract 
entities revealed by logical analysis ; Berkeley brings 
several weapons, but his chief is the subjectivity of 
sense-data, their dependence upon the organization and 
point of view of the spectator ; while modern physics, 
on the basis of sensible evidence itself, maintains a 
mad dance of electrons which have, superficiaJly ( at 
least, very little resemblance to the immediate objects 
of sight or touch. 

Every one of these lines of attack raises vital and 
interesting problems. 

The mystic, so long as he merely reports a positive 
revelation, cannot be refuted; but when he denies 


reality to objects of sense, he may be questioned as to 
what he means by " reality," and may be asked how 
their unreality follows from the supposed reality of his 
super-sensible world. In answering these questions, 
he is led to a logic which merges into that of 
and Plato and the idealist tradition. 

The logic of the idealist tradition has gradually grown 
very complex and very abstruse, as may be seen from 
the Bradleian sample considered in our first lecture. 
If we attempted to deal fully with this logic, we should 
not have time to reach any other aspect of our subject ; 
we will therefore, while acknowledging that it deserves 
a long discussion, pass by its central doctrines with 
only such occasional criticism as may serve to exemplify 
other topics, and concentrate our attention on such 
matters as its objections to the continuity of motion 
and the infinity of space and time objections which 
have been fully answered by modern mathematicians 
in a manner constituting an abiding triumph for the 
method of logical analysis in philosophy. These 
objections and the modern answers to them will occupy 
our fifth, sixth, and seventh lectures. 

Berkeley's attack, as reinforced by the physiology of 
the sense-organs and nerves and brain, is very powerful. 
I think it must be admitted as probable that the imme- 
diate objects of sense depend for their existence upon 
physiological conditions in ourselves, and that, for 
example, the coloured surfaces which we see cease to 
exist when we shut our eyes. But it would be a mistake 
to infer that they are dependent upon mind, not real 
while we see them, or not the sole basis for our know- 
ledge of the external world. This line of argument 
will be developed in the present lecture. 

The discrepancy between the world of physics and 
the world of sense, which we shall consider in our 


fourth lecture, will be found to be more apparent 
than real, and it will be shown that whatever there 
is reason to believe in physics can probably be inter- 
preted consistently with the reality of sense-data. 

The instrument of discovery throughout is modern 
logic, a very different science from the logic of the 
text-books and also from the logic of idealism. Our 
second lecture has given a short account of modern 
logic and of its points of divergence from the various 
traditional kinds of logic. 

In our last lecture, after a discussion of causality 
and free will, we shall try to reach a general account 
of the logical-analytic method of scientific philosophy, 
and a tentative estimate of the hopes of philosophical 
progress which it allows us to entertain. 

In this lecture, I wish to apply the logical-analytic 
method to one of the oldest problems of philosophy, 
namely, the problem of our knowledge of the external 
world. What I have to say on this problem does not 
amount to an answer of a definite and dogmatic kind ; 
it amounts only to an analysis and statement of the 
questions involved, with an indication of the directions 
in which evidence may be sought. But although not 
yet a definite solution, what can be said at present 
seems to me to throw a completely new light on the 
problem, and to be indispensable, not only in seeking 
the answer, but also in the preliminary question as 
to what parts of our problem may possibly have an 
ascertainable answer. 

In every philosophical problem, our investigation 
starts from what may be called " data," by which I 
mean matters of common knowledge, vague, complex, 
inexact, as common knowledge always is, but yet 
somehow commanding our assent as on the whole and 
in some interpretation pretty certainly true. In the 


case of our present problem, the common knowledge 
involved is of various kinds. There is first our acquain- 
tance with particular objects of daily life furniture, 
houses, towns, other people, and so on. Then there 
is the extension of such particular knowledge to par- 
ticular things outside our personal experience, through 
history and geography, newspapers, etc. And lastly, 
there is the systematization of all this knowledge of 
particulars by means of physical science, which derives 
immftngft persuasive force from its astonishing power 
of foretelling the future. We are quite willing to 
admit that there may be errors of detail in this know- 
ledge, but we believe them to be discoverable and 
corrigible by the methods which have given rise to our 
beliefs, and we do not, as practical men, entertain 
for a moment the hypothesis that the whole edifice 
may be built on insecure foundations. In the main, 
therefore, and without absolute dogmatism as to this 
or that special portion, we may accept this mass of 
common knowledge as affording data for our philo- 
sophical analysis. 

It may be said and this is an objection which must 
be met at the outset that it is the duty of the philo- 
sopher to call in question the admittedly fallible beliefs 
of daily life, and to replace them by something more 
solid and irrefragable. In a sense this is true, and in 
a sense it is effected in the course of analysis. But 
in another sense, and a very important one, it is quite 
impossible. While admitting that doubt is possible 
with regard to all our common knowledge, we must 
nevertheless accept that knowledge in the main if 
philosophy is to be possible at alL There is not any 
superfine brand of knowledge, obtainable by the 
philosopher, which can give us a standpoint from which 
to criticize the whole of the knowledge of daily life. 


The most that can be done is to examine and purify 
our common knowledge by an internal scrutiny, 
assuming the canons by which it has been obtained, 
and applying them with more care and with more 
precision. Philosophy cannot hoajsljolha^ing. achieved 
such adggrfeof certjdnty jthat itj^_have authority" 
' and the laiws'of 

sceptical in regard to every detail, is not sceptical as 
regards the whole. That is to say, its criticism of details 
will only be based upon their relation to other details, 
not upon some external criterion which can be applied 
to all the details equally. The reason for this absten- 
tion from a universal criticism is not any dogmatic 
confidence, but its exact opposite; it is not that 
common knowledge must be true, but that we possess 
no radically different kind of knowledge derived from 
some other source. Universal scepticism, though 
logically irrefutable, is practically barren ; it can only, 
therefore, give a certain flavour of hesitancy to our 
beliefs, and cannot be used to substitute other beliefs 
for them. 

Although data can only be criticized by other data, 
not by an outside standard, yet we may . distinguish 
different grades of certainty in the different kinds of 
common knowledge which we enumerated just now. 
What does not go beyond our own personal sensible 
acquaintance must be for us the most certain : the 
" evidence of the senses " is proverbially the least 
open to question. What depends on testimony, like 
the facts of history and geography which are learnt 
from books, has varying degrees of certainty according 
to the nature and extent of the testimony. Doubts 
as to the existence of Napoleon can only be maintained 
for a joke, whereas the historicity of Agamemnon is 


a legitimate subject of debate. In science, again, we 
find all grades of certainty short of the highest. The 
law of gravitation, at least as an approximate truth, 
has acquired by this time the same kind of certainty 
as the existence of Napoleon, whereas the latest specu- 
lations concerning the constitution of matter would 
be universally acknowledged to have as yet only a 
rather slight probability in their favour. These varying 
degrees of certainty attaching to different data may 
be regarded as themselves forming part of our data ; 
they, along with the other data, lie within the vague, 
complex, inexact body of knowledge which it is the 
business of the philosopher to analyse. 

The first thing that appears when we begin to analyse 
our common knowledge is that some of it is derivative, 
while some is primitive ; that is to say, there is some 
that we only believe because of something else from 
which it has been inferred in some sense, though not 
necessarily in a strict logical sense, while other parts 
are believed on their own account, without the support 
of any outside evidence. It is obvious that the senses 
give knowledge of the latter kind: the immediate 
facts perceived by sight or touch or hearing do not 
need to be proved by argument, but are completely 
self-evident. Psychologists, however, have made us 
aware that what is actually given in sense is much 
less than most people would naturally suppose, and 
that much of what at first sight seems to be given is 
really inferred. This applies especially in regard to our 
space-perceptions. For instance, we unconsciously infer 
the " real " size and shape of a visible object from its 
apparent size and shape, according to its distance and 
our point of view. When we hear a person speaking, 
our actual sensations usually miss a great deal of what 
he says, and we supply its place by unconscious 


inference ; in a foreign language, where this process 
is more difficult, we find ourselves apparently grown 
deaf, requiring, for example, to be much nearer the 
stage at a theatre than would be necessary in our 
own country. Thus the first step in the analysis of 
data, namely, the discovery of what is really given in 
sense, is full of difficulty. We will, however, not 
linger on this point ; so long as its existence is realized, 
the exact outcome does not make any very great difier- 
ence in our main problem. 

The next step in our analysis must be the con- 
sideration of how the derivative parts of our common 
knowledge arise. Here we become involved in a some- 
what puzzling entanglement of logic and psychology. 
Psychologically, a belief may be called derivative 
whenever it is caused by one or more other beliefs, 
or by some fact of sense which is not simply what the 
belief asserts. Derivative beliefs in this sense con- 
stantly arise without any process of logical inference, 
merely by association of ideas or some equally extra- 
logical process. From the expression of a man's face 
we judge as to what he is feeling : we say we see that 
he is angry, when in fact we only see a frown. We do 
not judge as to his state of mind by any logical process : 
the judgment grows up, often without our being able 
to say what physical mark of emotion we actually 
saw. In such a case, the knowledge is derivative 
psychologically ; but logically it is in a sense primitive, 
since it is not the result of any logical deduction. 
There may or may not be a possible deduction leading 
to the same result, but whether there is or not, we 
certainly do not employ it. If we call a belief " logically 
primitive" when it is not actually arrived at by a 
logical inference, then innumerable beliefs are logically 
primitive which psychologically are derivative. The 


separation of these two kinds of primitiveness is vitally 
important to our present discussion. 

When we reflect upon the beliefs which axe logically 
but not psychologically primitive, we find that, unless 
they can on reflection be deduced by a logical process 
from beliefs which are also psychologically primitive, 
our confidence in their truth tends to diminish the more 
we think about them. We naturally believe, for 
example, that tables and chairs, trees and mountains, 
are still there when we turn our backs upon them. I 
do not wish for a moment to maintain that this is 
certainly not the case, but I do maintain that the 
question whether it is the case is not to be settled off- 
hand on any supposed ground of obviousness. The 
belief that they persist is, in all men except a few 
philosophers, logically primitive, but it is not psycho- 
logically primitive; psychologically, it arises only 
through our having seen those tables and chairs, trees 
and mountains. As soon as the question is seriously 
raised whether, because we have seen them, we have a 
right to suppose that they are there still, we feel that 
some kind of argument must be produced, and that if 
none is forthcoming, our belief can be no more than 
a pious opinion. We do not feel this as regards the 
immediate objects of sense : there they are, and as 
far as their momentary existence is concerned, no 
further argument is required. There is accordingly 
more need of justifying our psychologically derivative 
beliefs than of justifying those that are primitive. 

We are thus led to a somewhat vague distinction 
between what we may call " hard " data and " soft " 
data. This distinction is a matter of degree, and must 
not be pressed ; but if not taken too seriously, it may 
help to make the situation clear. I mean by " hard " 
data those which resist the solvent influence of critical 


reflection, and by " soft " data those which, under the 
operation of this process, become to our minds more 
or less doubtful. The hardest of hard data are of two 
sorts : the particular facts of sense, and the general 
truths of logic. The more we reflect upon these, the 
more we realize exactly what they are, and exactly 
what a doubt concerning them really means, the more 
luminously certain do they become. Verbal doubt 
concerning even these is possible, but verbal doubt 
may occur when what is nominally being doubted is 
not really in our thoughts, and only words are actually 
present to our minds. Real doubt, in these two cases, 
would, I think, be pathological. At any rate, to me 
they seem quite certain, and I shall assume that you 
agree with me in this. Without this assumption, we 
are in danger of falling into that universal scepticism 
which, as we saw, is as barren as it is irrefutable. 
If we are to continue philosophizing, we must make 
our bow to the sceptical hypothesis, and, while 
admitting the elegant terseness of its philosophy, 
proceed to the consideration of other hypotheses 
which, though perhaps not certain, have at least as 
good a right to our respect as the hypothesis of the 

Applying our distinction of "hard" and "soft" 
data to psychologically derivative but logically primi- 
tive beliefs, we shall find that most, if not all, are to be 
classed as soft data. They may be found, on reflection, 
to be capable of logical proof, and they then again 
become believed, but no longer as data. As data, 
though entitled to a certain limited respect, they cannot 
be placed on a level with the facts of sense or the laws 
of logic. The kind of respect which they deserve 
seems to me such as to warrant us in hoping, though 
not too confidently, that the hard data may prove them 


to be at least probable. Also, if the hard data are 
found to throw no light whatever upon their truth or 
falsehood, we are justified, I think, in giving rather 
more weight to the hypothesis of their truth than to 
the hypothesis of their falsehood. For the present, 
however, let us confine ourselves to the hard data, 
with a view to discovering what sort of world can be 
constructed by their means alone. 

Our data now are primarily the facts of sense (i.e. 
of our own sense-data) and the laws of logic. But even 
the severest scrutiny will allow some additions to this 
slender stock. Some facts of memory especially 
of recent memory seem to have the highest degree of 
certainty. Some introspective facts are as certain as 
any facts of sense. And facts of sense themselves must, 
for our present purposes, be interpreted with a certain 
latitude. Spatial and temporal relations must some- 
times be included, for example in the case of a swift 
motion falling wholly within the specious present. 
And some facts of comparison, such as the likeness 
or unlikeness of two shades of colour, are certainly 
to be included among hard data. Also we must remem- 
ber that the distinction of hard and soft data is psycho- 
logical and subjective, so that, if there are other 
minds than our own which at our present stage must 
be held doubtful the catalogue of hard data may be 
different for them from what it is for us. 

Certain common beliefs are undoubtedly excluded 
from hard data. Such is the belief which led us to 
introduce the distinction, namely, that sensible objects 
in general persist when we are not perceiving them. 
Such also is the belief in other people's minds : this 
belief is psychologically derivative from our perception 
of their bodies, and is fdt to demand logical justifica- 
tion as soon as we become aware of its derivativeness. 


Belief in what is reported by the testimony of others, 
including all that we learn from books, is of course 
involved in the doubt as to whether other people 
have minds at all. Thus the world from which our 
reconstruction is to begin is very fragmentary. The 
best we can say for it is that it is slightly more extensive 
than the world at which Descartes arrived by a similar 
process, since that world contained nothing except 
himself and his thoughts. 

We are now in a position to understand and state 
the problem of our knowledge of the external world, 
and to remove various misunderstandings which have 
obscured the meaning of the problem. The problem 
really is : Can the existence of anything other 
than our own hard data be inferred from the 
existence of those data? But before considering 
this problem, let us briefly consider what the problem 
is not. 

When we speak of the " external " world in this 
discussion, we must not mean " spatially external," 
unless " space " is interpreted in a peculiar and recon- 
dite manner. The immediate objects of sight, the 
coloured surfaces which make up the visible world, 
are spatially external in the natural meaning of this 
phrase. We fed them to be " there " as opposed to 
" here " ; without making any assumption of an 
existence other than hard data, we can more or less 
estimate the distance of a coloured surface. It seems 
probable that distances, provided they are not too 
great, are actually given more or less roughly in sight ; 
but whether this is the case or not, ordinary distances 
can certainly be estimated approximately by means 
of the data of sense alone. The immediately given 
world is spatial, and is further not wholly contained 
within our own bodies, at least in the obvious sense. 


Thus our knowledge of what is external in this sense 
is not open to doubt. 

Another form in which the question is often put is : 
" Can we know of the existence of any reality which is 
independent of ourselves ? " This form of the question 
suffers from the ambiguity of the two words " inde- 
pendent " and " self." To take the Self first : the 
question as to what is to be reckoned part of the Self 
and what is not, is a very difficult one. Among many 
other things which we may mean by the Self, two may 
be selected as specially important, namely (i) the bare 
subject which thinks and is aware of objects, (2) the 
whole assemblage of things that would necessarily 
cease to exist if our lives came to an end. The bare 
subject, if it exists at all, is an inference, and is not 
part of the data; therefore, this meaning of Self 
may be ignored in our present inquiry. The second 
meaning is difficult to make precise, since we hardly 
know what things depend upon our lives for their 
existence. And in this form, the definition of Self 
introduces the word " depend," which raises the same 
questions as are raised by the word " independent." 
Let us therefore take up the word " independent," 
and return to the Self later. 

When we say that one thing is " independent " 
of another, we may mean either that it is logically 
possible for the one to exist without the other, or that 
there is no causal relation between the two such that 
the one only occurs as the effect of the other. The 
only way, so far as I know, in which one thing can be 
logically dependent upon another is when the other 
is part of the one. The existence of a book, for example, 
is logically dependent upon that of its pages : without 
the pages there would be no book. Thus in this sense 
the question, " Can we know of the existence of any 



reality which is independent of ourselves ? " reduces 
to the question, " Can we know of the existence of 
any reality of which our Self is not part ? " In this 
form, the question brings us back to the problem of 
defining the Self ; but I think, however the Self may be 
defined, even when it is taken as the bare subject, 
it cannot be supposed to be part of the immediate 
object of sense ; thus in this form of the question we 
must admit that we can know of the existence of 
realities independent of ourselves. 

The question of causal dependence is much more 
difficult. To know that one kind of thing is causally 
independent of another, we must know that it actually 
occurs without the other. Now it is fairly obvious 
that, whatever legitimate meaning we give to the Self, 
our thoughts and feelings are causally dependent 
upon ourselves, i.e. do not occur when there is no 
Self for them to belong to. But in the case of objects 
of sense this is not obvious ; indeed, as we saw, the 
common-sense view is that such objects persist in the 
absence of any percipient. If this is the case, then they 
are causally independent of ourselves; if not, not. 
Thus in this form the question reduces to the question 
whether we can know that objects of sense, or any other 
objects not our own thoughts and feelings, exist at 
times when we are not perceiving them. This form, 
in which the difficult word " independent " no longer 
occurs, is the form in which we stated the problem a 
minute ago. 

Our question in the above form raises two distinct 
problems, which it is important to keep separate. 
First, can we know that objects of sense, or very 
similar objects, exist at times when we are not perceiv- 
ing them ? Secondly, if this cannot be known, can 
we know that other objects, inferable from objects 


of sense but not necessarily resembling them, exist 
either when we are perceiving the objects of sense 
or at any other time? This latter problem arises 
in philosophy as the problem of the " thing in itself," 
and in science as the problem of matter as assumed 
in physics. We will consider this latter problem 

According to some authors among whom I was 
formerly included it is necessary to distinguish 
between a sensation, which is a mental event, and its 
object, which is a patch of colour or a noise or what 
not. If this distinction is made, the object of the 
sensation is called a " sense-datum " or a " sensible 
object." Nothing in the problems to be discussed in 
this book depends upon the question whether this 
distinction is valid or not. If it is not valid, the sensa- 
tion and the sense-datum are identical If it is valid, 
it is the sense-datum which concerns us in this book, 
not the sensation. For reasons explained in The Analy- 
sis of Mind (e.g. p. 141 ff .) I have come to regard the 
distinction as not valid, and to consider the sense- 
datum identical with the sensation. But it will not 
be necessary to assume the correctness of this view 
in what follows. 

When I speak of a " sensible object," it must be 
understood that I do not mean such a thing as a table, 
which is both visible and tangible, can be seen by 
many people at once, and is more or less permanent. 
What I mean is just that patch of colour which is 
momentarily seen when we look at the table, or just 
that particular hardness which is felt when we press 
it, or just that particular sound which is heard when 
we rap it. Both the thing-in-itself of philosophy 
and the matter of physics present themselves as causes 
of the sensible object as much as of the sensation 


(if these are distinct). What are the common grounds 
for this opinion ? 

In each case, I think, the opinion has resulted from 
the combination of a belief that something which can 
persist independently of our consciousness makes itself 
known in sensation, with the fact that our sensations 
often change in ways which seem to depend upon us 
rather than upon anything which would be supposed 
to persist independently of us. At first, we believe 
unreflectingly that everything is as it seems to be, 
and that, if we shut our eyes, the objects we had been 
seeing remain as they were though we no longer see 
them. But there are arguments against this view, 
which have generally been thought conclusive. It 
is extraordinarily difficult to see just what the 
arguments prove; but if we are to make any 
progress with the problem of the external world, 
we must try to make up our minds as to these 

A table viewed from one place presents a different 
appearance from that which it presents from another 
place. This is the language of common sense, but 
this language already assumes that there is a real table 
of which we see the appearances. Let us try to state 
what is known in terms of sensible objects alone, 
without any element of hypothesis. We find that as we 
walk round the table, we perceive a series of gradually 
changing visible objects. But in speaking of " walking 
round the table," we have still retained the hypothesis 
that there is a single table connected with all the 
appearances. What we ought to say is that, while 
we have those muscular and other sensations which 
make us say we are walking, our visual sensations 
change in a continuous way, so that, for example, 
a striking patch of colour is not suddenly replaced by 


something wholly different, but is replaced by an 
insensible gradation of slightly different colours with 
slightly different shapes. This is what we really know 
by experience, when we have freed our minds from 
the assumption of permanent " things " with changing 
appearances. What is really known is a correlation of 
muscular and other bodily sensations with changes 
in visual sensations. 

But walking round the table is not the only way of 
altering its appearance. We can shut one eye, or put 
on blue spectacles, or look through a microscope. 
All these operations, in various ways, alter the visual 
appearance which we call that of the table. More 
distant objects will also alter their .appearance if (as 
we say) the state of the atmosphere changes if there 
is fog or rain or sunshine. Physiological changes also 
alter the appearances of things. If we assume the 
world of common sense, all these changes, including 
those attributed to physiological causes, are changes 
in the intervening medium. It is not quite so easy as in 
the former case to reduce this set of facts to a form 
in which nothing is assumed beyond sensible objects. 
Anything intervening between ourselves and what we 
see must be invisible : our view in every direction is 
bounded by the nearest visible object. It might 
be objected that a dirty pane of glass, for example, 
is visible, although we can see things through it. But 
in this case we really see a spotted patchwork : the 
dirtier specks in the glass are visible, while the cleaner 
parts are invisible and allow us to see what is beyond. 
Thus the discovery that the intervening medium affects 
the appearances of things cannot be made by means of 
the sense of sight alone. 

Let us take the case of the blue spectacles, which is 
the simplest, but may serve as a type for the others. 


The frame of the spectacles is of course visible, but the 
blue glass, if it is dean, is not visible. The blueness, 
which we say is in the glass, appears as being in the 
objects seen through the glass. The glass itself is known 
by means of the sense of touch. In order to know that 
it is between us and the objects seen through it, we 
must know how to correlate the space of touch with 
the space of sight. This correlation itself, when stated 
in terms of the data of sense alone, is by no means a 
simple matter. But it presents no difficulties of 
principle, and may therefore be supposed accomplished. 
When it has been accomplished, it becomes possible to 
attach a meaning to the statement that the blue glass, 
which we can touch, is between us and the object seen, 
as we say, " through " it. 

But we have still not reduced our statement com* 
pletdy to what is actually given in sense. We have 
fallen into the assumption that the object of which 
we are conscious when we touch the blue spectacles 
still exists after we have ceased to touch them. So 
long as we are touching them, nothing except our 
finger can be seen through the part touched, which is 
the only part where we immediately know that there 
is something. If we are to account for the blue appear- 
ance of objects other than the spectacles, when seen 
through them, it might seem as if we must assume that 
the spectacles still exist when we are not touching 
them ; and if this assumption really is necessary, our 
main problem is answered : we have means of knowing 
of the present existence of objects not given in sense, 
though of the same kind as objects formerly given 
in sense. 

It may be questioned, however, whether this assump- 
tion is actually unavoidable, though it is unquestionably 
the most natural one to make. We may say that the 


object of which we become aware when we touch the 
spectacles continues to have effects afterwards, though 
perhaps it no longer exists. In this view, the supposed 
continued existence of sensible objects after they have 
ceased to be sensible will be a fallacious inference 
from the fact that they still have effects. It is often 
supposed that nothing which has ceased to exist can 
continue to have effects, but this is a mere preju- 
dice, due to a wrong conception of causality. We 
cannot, therefore, dismiss our present hypothesis 
on the ground of a priori impossibility, but must 
examine further whether it can really account for 
the facts. 

It may be said that our hypothesis is useless in the 
case when the blue glass is never touched at all. How, 
in that case, are we to account for the blue appearance 
of objects ? And more generally, what are we to make 
of the hypothetical sensations of touch which we 
associate with untouched visible objects, which we 
know would be verified if we chose, though in fact we 
do not verify them ? Must not these be attributed to 
permanent possession, by the objects, of the properties 
which touch would reveal ? 

Let us consider the more general question first. 
Experience has taught us that where we see certain 
kinds of coloured surfaces we can, by touch, obtain 
certain expected sensations of hardness or softness, 
tactile shape, and so on. This leads us to believe that 
what is seen is usually tangible, and that it has, whether 
we touch it or not, the hardness or softness which we 
should expect to fed if we touched it. But the mere 
fact that we are able to infer what our tactile sensations 
would be shows that it is not logically necessary to 
assume tactile qualities before they are felt. All that 
is really known is that the visual appearance in question 


together with touch, will lead to certain sensations, 
which can necessarily be determined in terms of the 
visual appearance, since otherwise they could not be 
inferred from it. 

We can now give a statement of the experienced 
facts concerning the blue spectacles, which will supply 
an interpretation of common-sense beliefs without 
assuming anything beyond the existence of sensible 
objects at the times when they are sensible. By 
experience of the correlation of touch and sight sensa- 
tions, we become able to associate a certain place in 
touch-space with a certain corresponding place in 
sight-space. Sometimes, namely in the case of trans- 
parent things, we find that there is a tangible object 
in a touch-place without there being any visible object 
in the corresponding sight-place. But in such a case 
as that of the blue spectacles, we find that whatever 
object is visible beyond the empty sight-place in the 
same line of sight has a different colour from what it 
has when there is no tangible object in the intervening 
touch-place ; and as we move the tangible object in 
touch-space, the blue patch moves in sight-space. If 
now we find a blue patch moving in this way in sight- 
space, when we have no sensible experience of an 
intervening tangible object, we nevertheless infer that, 
if we put our hand at a certain place in touch-space, 
we should experience a certain touch-sensation. If we 
are to avoid non-sensible objects, this must be taken 
as the whole of our meaning when we say that the 
blue spectacles are in a certain place, though we have 
not touched them, and have only seen other things 
rendered blue by their interposition. 

I think it may be laid down quite generally that, 
in so far as physics or common sense is verifiable, it 
must be capable of interpretation in terms of actual 


sense-data alone. The reason for this is simple. 
Verification consists always in the occurrence of an 
expected sense-datum. Astronomers tell us there 
will be an eclipse of the moon : we look at the moon, 
and find the earth's shadow biting into it, that is to 
say, we see an appearance quite different from that 
of the usual full moon. Now if an expected sense- 
datum constitutes a verification, what was asserted 
must have been about sense-data; or, at any rate, 
if part of what was asserted was not about sense-data, 
then only the other part has been verified. There is 
in fact a certain regularity or conformity to law about 
the occurrence of sense-data, but the sense-data that 
occur at one time are often causally connected with 
those that occur at quite other times, and not, or 
at least not very closely, with those that occur at 
neighbouring times. If I look at the moon and imme- 
diately afterwards hear a train coming, there is no very 
close causal connection between my two sense-data ; 
but if I look at the moon on two nights a week apart, 
there is a very dose causal connection between the two 
sense-data. The simplest, or at least the easiest, 
statement of the connection is obtained by imagining 
a " real " moon which goes on whether I look at it 
or not, providing a series of possible sense-data of 
which only those are actual which belongs to moments 
when I choose to look at the moon. 

But the degree of verification obtainable in this way 
is very small. It must be remembered that, at our 
present level of doubt, we are not at liberty to accept 
testimony. When we hear certain noises, which are 
those we should utter if we wished to express a certain 
thought, we assume that that thought, or one very 
like it, has been in another mind, and has given rise 
to the expression which we hear. If at the same time 


we see a body resembling our own, moving its lips 
as we move ours when we speak, we cannot resist the 
belief that it is alive, and that the feelings inside it 
continue when we are not looking at it. When we 
see our friend drop a weight upon his toe, and hear 
him say what we should say in similar circumstances, 
the phenomena can no doubt be explained without 
assuming that he is anything but a series of shapes 
and noises seen and heard by us, but practically no 
man is so infected with philosophy as not to be quite 
certain that his friend has felt the same kind of pain as 
he himself would feel. We will consider the legitimacy 
of this belief presently ; for the moment, I only wish 
to point out that it needs the same kind of justification 
as our belief that the moon exists when we do not see 
it, and that, without it, testimony heard or read is 
reduced to noises and shapes, and cannot be regarded 
as evidence of the facts which it reports. The verifica- 
tion of physics which is possible at our present level 
is, therefore, only that degree of verification which is 
possible by one man's unaided observations, which 
will not carry us very far towards the establishment 
of a whole science. 

Before proceeding further, let us summarize the 
argument so far as it has gone. The problem is : " Can 
the existence of anything other than our own hard 
data be inferred from these data ? " It is a mistake 
to state the problem in the form : " Can we know of 
the existence of anything other than ourselves and 
our states ? " or : " Can we know of the existence of 
anything independent of ourselves ? " because of the 
extreme difficulty of defining "self" and "inde- 
pendent " precisely. The felt passivity of sensation 
is irrelevant, since, even if it proved anything, it 
could only prove that sensations are caused by sensible 


objects. The natural naive belief is that things seen 
persist, when unseen, exactly or approximately as 
they appeared when seen ; but this belief tends to be 
dispelled by the fact that what common sense regards 
as the appearance of one object changes with what 
common sense regards as changes in the point of 
view and in the intervening medium, including in the 
latter our own sense-organs and nerves and brain. 
This fact, as just stated, assumes, however, the common- 
sense world of stable objects which it professes to call 
in question ; hence, before we can discover its precise 
bearing on our problem, we must find a way of stating 
it which does not involve any of the assumptions 
which it is designed to render doubtful. What we 
then find, as the bare outcome of experience, is that 
gradual changes in certain sense-data are correlated 
with gradual changes in certain others, or (in the 
case of bodily motions) with the other sense-data 

The assumption that sensible objects persist after 
they have ceased to be sensible for example, that 
the hardness of a visible body, which has been dis- 
covered by touch, continues when the body is no longer 
touched may be replaced by the statement that the 
effects of sensible objects persist, i.e. that what happens 
now can only be accounted for, in many cases, by 
taking account of what happened at an earlier time. 
Everything that one man, by his own personal experi- 
ence, can verify in the account of the world given by 
common sense and physics, will be explicable by some 
such means, since verification consists merely in the 
occurrence of an expected sense-datum. But what 
depends upon testimony, whether heard or read, cannot 
be explained in this way, since testimony depends 
upon the existence of minds other than our own, and 


thus requires a knowledge of something not given in 
sense. But before examining the question of our 
knowledge of other minds, let us return to the question 
of the thing-in-itself, namely, to the theory that what 
exists at times when we axe not perceiving a given 
sensible object is something quite unlike that object, 
something which, together with us and our sense- 
organs, causes our sensations, but is never itself given 
in sensation. 

The thing-in-itself, when we start from common- 
sense assumptions, is a fairly natural outcome of the 
difficulties due to the changing appearances of what 
is supposed to be one object. It is supposed that the 
table (for example) causes our sense-data of sight and 
touch, but must, since these are altered by the point 
of view and the intervening medium, be quite different 
from the sense-data to which it gives rise. The objection 
to this theory, I think, lies in its failure to realize 
the radical nature of the reconstruction demanded by 
the difficulties to which it points. We cannot speak 
legitimately of changes in the point of view and the 
intervening medium until we have already constructed 
some world more stable than that of momentary 
sensation. Our discussion of the blue spectacles and 
the walk round the table has, I hope, made this dear. 
But what remains far from dear is the nature of the 
reconstruction required. 

Although we cannot rest content with the above 
theory, in the terms in which it is stated, we must 
neverthdess treat it with a certain respect, for it is 
in outline the theory upon which physical science and 
physiology are built, and it must, therefore, be suscep- 
tible of a true interpretation. Let us see how this is 
to be done. 

The first thing to realize is that there are no such 


things as " illusions of sense." Objects of sense, even 
when they occur in dreams, are the most indubitably 
real objects known to us. What, then, makes us call 
them unreal in dreams ? Merely the unusual nature 
of their connection with other objects of sense. I dream 
that I am in America, but I wake up and find myself 
in England without those intervening days on the 
Atlantic which, alas 1 are inseparably connected with 
a "real" visit to America. Objects of sense are 
called " real " when they have the kind of connection 
with other objects of sense which experience has led us 
to regard as normal ; when they fail in this, they are 
called " illusions." But what is illusory is only the 
inferences to which they give rise ; in themselves, they 
are every bit as real as the objects of waking life. 
And conversely, the sensible objects of waking life 
must not be expected to have any more intrinsic reality 
than those of dreams. Dreams and waking life, in our 
first efforts at construction, must be treated with equal 
respect ; it is only by some reality not merely sensible 
that dreams can be condemned. 

Accepting the indubitable momentary reality of 
objects of sense, the next thing to notice is the 
confusion underlying objections derived from their 
changeableness. As we walk round the table, its 
aspect changes ; but it is thought impossible to maintain 
either that the table changes, or that its various 
aspects can all " really " exist in the same place. If 
we press one eyeball, we shall see two tables ; but it 
is thought preposterous to maintain that there are 
"really" two tables. Such arguments, however, 
seem to involve the assumption that there can be 
something more real than objects of sense. If we 
see two tables, then there are two visual tables. It 
is perfectly true that, at the same moment, we may 


discover by touch that there is only one tactile table. 
This makes us declare the two visual tables an illusion, 
because usually one visual object corresponds to one 
tactile object. But all that we are warranted in saying 
is that, in this case, the manner of correlation of touch 
and sight is unusual. Again, when the aspect of the 
table changes as we walk round it, and we are told 
there cannot be so many different aspects in the same 
place, the answer is simple : what does the critic of 
the table mean by " the same place " ? The use of 
such a phrase presupposes that all our difficulties have 
been solved ; as yet, we have no right to speak of a 
"place" except with reference to one given set of 
momentary sense-data. When all are changed by a 
bodily movement, no place remains the same as it 
was. Thus the difficulty, if it exists, has at least not 
been rightly stated. 

We will now make a new start, adopting a different 
method. Instead of inquiring what is the minimum of 
assumption by which we can explain the world of sense, 
we will, in order to have a model hypothesis as a 
help for the imagination, construct one possible 
(not necessary) explanation of the facts. It may 
perhaps then be possible to pare away what is 
superfluous in our hypothesis, leaving a residue 
which may be regarded as the abstract answer to our 

Let us imagine that each mind looks out upon the 
world, as in Leibniz's monadology, from a point of 
view peculiar to itself ; and for the sake of simplicity 
let us confine ourselves to the sense of sight, ignoring 
minds which are devoid of this sense. Each mind sees 
at each moment an immensely complex three-dimen- 
sional world ; but there is absolutely nothing which 
is seen by two minds simultaneously. When we say 


that two people see the same thing, we always find that, 
owing to difference of point of view, there are differences, 
however slight, between their immediate sensible 
objects. (I am here assuming the validity of testimony 
but as we are only constructing a possible theory, that 
is a legitimate assumption.) The three-dimensional 
world seen by one mind therefore contains no place 
in common with that seen by another, for places can 
only be constituted by the things in or around them. 
Hence we may suppose, in spite of the differences 
between the different worlds, that each exists entire 
exactly as it is perceived, and might be exactly as it 
is even if it were not perceived. We may further 
suppose that there are an infinite number of such 
worlds which are in fact unperceived. If two men are 
sitting in a room, two somewhat similar worlds are 
perceived by them ; if a third man enters and sits 
between them, a third world, intermediate between 
the two previous worlds, begins to be perceived. 
It is true that we cannot reasonably suppose just this 
world to have existed before, because it is conditioned 
by the sense-organs, nerves, and brain of the newly 
arrived man; but we can reasonably suppose that 
some aspect of the universe existed from that point of 
view, though no one was perceiving it. The system 
consisting of all views of the universe, perceived and 
unperceived, I shall call the system of "perspectives" ; 
I shall confine the expression " private worlds " to 
such views of the universe as are actually perceived. 
Thus a " private world " is a perceived "perspective " 
but there may be any number of unperceived per- 

Two men are sometimes found to perceive very 
similar perspectives, so similar that they can use the 
same words to describe them. They say they see 


the same table, because the differences between the two 
tables they see are slight and not practically important. 
Thus it is possible, sometimes, to establish a correlation 
by similarity between a great many of the things of 
one perspective, and a great many of the things of 
another. In case the similarity is very great, we say 
the points of view of the two perspectives are near 
together in space-; but this space in which they are 
near together is totally different from the spaces 
inside the two perspectives. It is a relation between the 
perspectives, and is not in either of them ; no one can 
perceive it, and if it is to be known it can be only by 
inference. Between two perceived perspectives which 
are similar, we can imagine a whole series of other 
perspectives, some at least unperceived, and such 
that between any two, however similar, there are others 
still more similar. In this way the space which consists 
of relations between perspectives can be rendered 
continuous, and (if we choose) three-dimensional. 

We can now define the momentary common-sense 
"thing," as opposed to its momentary appearances. 
By the similarity of neighbouring perspectives, many 
objects in the one can be correlated with objects in 
the other, namely with the similar objects. Given an 
object in one perspective, form the system of all the 
objects correlated with it in all the perspectives ; that 
system may be identified with the momentary com- 
mon-sense " thing." Thus an aspect of a "thing" is a 
member of the system of aspects which is the " thing " 
at that moment. (The correlation of the times of 
different perspectives raises certain complications, of 
the kind considered in the theory of relativity ; but 
we may ignore these at present.) All the aspects of a 
thing are real, whereas the thing is a merely logical 
construction. It has, however, the merit of being 


neutral as between different points of view, and of 
being visible to more than one person, in the only 
sense in which it can ever be visible, namely, in the 
sense that each sees one of its aspects. 

It will be observed that, while each perspective 
contains its own space, there is only one space in which 
the perspectives themselves axe the dements. There 
axe as many private spaces as there are perspectives ; 
there are therefore at least as many as there are per- 
cipients, and there may be any number of others which 
have a merely material existence and are not seen by 
anyone. But there is only one perspective-space, 
whose elements are single perspectives, each with 
its own private space. We have now to explain 
how the private space of a single perspective is cor- 
related with part of the one all-embracing perspective 

Perspective space is the system of " points of view " 
of private spaces (perspectives), or, since " points of 
view " have not been defined, we may say it is the 
system of the private spaces themselves. These 
private spaces will each count as one point, or at any 
rate as one element, in perspective space. They are 
ordered by means of their similarities. Suppose, for 
example, that we start from one which contains the 
appearance of a circular disc, such as would be called 
a penny, and suppose this appearance, in the perspec- 
tive in question, is circular, not elliptic. We can then 
f orm a whole series of perspectives containing a gradu- 
ated series of circular aspects of varying sizes : for this 
purpose we only have to move (as we say) towards 
the penny or away from it. The perspectives in which 
the penny looks circular will be said to lie on a straight 
line in perspective space, and their order on this line 
will be that of the sizes of the circular aspects. More- 



over though, this statement must be noticed and 
subsequently examined the perspectives in which the 
penny looks big will be said to be nearer to the penny 
than those in which it looks small. It is to be remarked 
also that any other " thing " than our penny might 
have been chosen to define the relations of our per- 
spectives in perspective space, and that experience 
shows that the same spatial order of perspectives 
would have resulted. 

In order to explain the correlation of private spaces 
with perspective space, we have first to explain what 
is meant by " the place (in perspective space) where 
a thing is." For this purpose, let us again consider 
the penny which appears in many perspectives. We 
formed a straight line of perspectives in which the penny 
looked circular, and we agreed that those in which it 
looked larger were to be considered as nearer to the 
penny. We can form another straight line of perspec- 
tives in which the penny is seen end-on and looks 
like a straight line of a certain thickness. These two 
lines will meet in a certain place in perspective space, 
i.e. in a certain perspective, which may be defined as 
" the place (in perspective space) where the penny is." 
It is true that, in order to prolong our lines until they 
reach this place, we shall have to make use of other 
things besides the penny, because, so far as experience 
goes, the penny ceases to present any appearance 
after we have come so near to it that it touches the 
eye. But this raises no real difficulty, because the 
spacial order of perspectives is found empirically 
to be independent of the particular " things " chosen 
for defining the order. We can, for example, remove 
our penny and prolong each of our two straight lines 
up to their intersection by placing other pennies 
further off in such a way that the aspects of the one are 


circular where those of our original penny were circular, 
and the aspects of the other are straight where those 
of our original penny were straight. There will then be 
just one perspective in which one of the new pennies 
looks circular and the other straight. This will be, by 
definition, the place where the original penny was in 
perspective space. 

The above is, of course, only a first rough sketch of 
the way in which our definition is to be reached. It 
neglects the size of the penny, and it assumes that we 
can remove the penny without being disturbed by any 
simultaneous changes in the positions of other things. 
But it is plain that such niceties cannot affect the 
principle, and can only introduce complications in 
its application. 

Having now defined the perspective, which is the 
place where a given thing is, we can understand what 
is meant by saying that the perspectives in which 
a think looks large are nearer to the things than 
those in which it looks small: they are, in fact, 
nearer to the perspective which is the place where the 
thing is. 

We can now also explain the correlation between a 
private space and parts of perspective space. If there 
is an aspect of a given thing in a certain private space, 
then we correlate the place where this aspect is in the 
private space with the place where the thing is in 
perspective space. 

We may define " here " as the place, in perspective 
space, which is occupied by our private world. Thus 
we can now understand what is meant by speaking of 
a thing as near to or far from " here." A thing is near 
to " here " if the place where it is is near to my private 
world. We can also understand what is meant by saying 
that our private world is inside our head ; for our 


private world is a place in perspective space, and may 
be part of the place where our head is. 

It will be observed that two places in perspective 
space are associated with every aspect of a thing: 
namely, the place where the thing is, and the place 
which is the perspective of which the aspect in question 
forms part. Every aspect of a thing is a member 
of two different classes of aspects, namely : (i) the 
various aspects of the thing, of which at most one 
appears in any given perspective ; (2) the perspective 
of which the given aspect is a member, i.e. that in which 
the thing has the given aspect. The physicist naturally 
classifies aspects in the first way, the psychologist in 
the second. The two places associated with a single 
aspect correspond to the two ways of classifying it. 
We may distinguish the two places as that at which, 
and that from which, the aspect appears. The " place 
at which " is the place of the thing to which the aspect 
belongs ; the " place from which " is the place of the 
perspective to which the aspect belongs. 

Let us now endeavour to state the fact that the aspect 
which a thing presents at a given place is affected by 
the intervening medium. The aspects of a thing in 
different perspectives are to be conceived as spreading 
outwards from the place where the thing is, and 
undergoing various changes as they get further away 
from this place. The laws according to which they 
change cannot be stated if we only take account of 
the aspects that are near the thing, but require that 
we should also take account of the things that are 
at the places from which these aspects appear. This 
empirical fact can, therefore, be interpreted in terms of 
our construction. 

We have now constructed a largely hypothetical 
picture of the world, which contains and places the 


experienced facts, including those derived from testi- 
mony. The world we have constructed can, with a 
certain amount of trouble, be used to interpret the 
crude facts of sense, the facts of physics, and the facts 
of physiology. It is therefore a world which may 
be actual. It fits the facts, and there is no empirical 
evidence against it ; it also is free from logical im- 
possibilities. But have we any good reason to suppose 
that it is real ? This brings us back to our original 
problem, as to the grounds for believing in the existence 
of anything outside my private world. What we have 
derived from our hypothetical construction is that there 
are no grounds against the truth of this belief, but we 
have not derived any positive grounds in its favour. 
We will resume this inquiry by taking up again the 
question of testimony and the evidence for the existence 
of other minds. 

It must be conceded to begin with that the argument 
in favour of the existence of other people's minds 
cannot be conclusive. A phantasm of our dreams will 
appear to have a mind a mind to be annoying, as 
a rule. It will give unexpected answers, refuse to con- 
form to our desires, and show all those other signs 
of intelligence to which we are accustomed in the 
acquaintances of our waking hours. And yet, when 
we are awake, we do not believe that the phantasm 
was, like the appearances of people in waking life, 
representative of a private world to which we have 
no direct access. If we are to believe this of the people 
we meet when we are awake, it must be on some ground 
short of demonstration, since it is obviously possible 
that what we call waking life may be only an unusually 
persistent and recurrent nightmare. It may be that 
our imagination brings forth all that other people 
seem to say to us, all that we read in books, all the 


daily, weekly, monthly, and quarterly journals that 
distract our thoughts, all the advertisements of soap 
and all the speeches of politicians. This may be true, 
since it cannot be shown to be false, yet no one can 
really believe it. Is there any logical ground for regard- 
ing this possibility as improbable ? Or is there nothing 
beyond habit and prejudice ? 

The minds of other people are among our data, in 
the very wide sense in which we used the word at first. 
That is to say, when we first begin to reflect, we find 
ourselves already believing in them, not because of 
any argument, but because the belief is natural to us. 
It is, however, a psychologically derivative belief, 
since it results from observation of people's bodies ; 
and along with other such beliefs, it does not belong 
to the hardest of hard data, but becomes, under the 
influence of philosophic reflection, just sufficiently 
questionable to make us desire some argument con- 
necting it with the facts of sense. 

The obvious argument is, of course, derived from 
analogy. Other people's bodies behave as ours do when 
we have certain thoughts and feelings; hence, by 
analogy, it is natural to suppose that such behaviour 
is connected with thoughts and feelings like our own. 
Someone says " Look out 1 " and we find we are on the 
point of being killed by a motor-car; we therefore 
attribute the words we heard to the person in question 
having seen the motor-car first, in which case there are 
existing things of which we are not directly conscious. 
But this whole scene, with our inference, may occur 
in a dream, in which case the inference is generally 
considered to be mistaken. Is there anything to make 
the argument from analogy more cogent when we are 
(as we think) awake ? 

The analogy in waking life is only to be preferred to 


that in dreams on the ground of its greater extent and 
consistency. If a man were to dream every night about 
a set of people whom he never met by day, who had 
consistent characters and grew older with the lapse 
of years, he might, like the man in Calderon's play, 
find it difficult to decide which was the dream-world 
and which was the so-called ' * real " world. It is only 
the failure of our dreams to form a consistent whole 
either with each other or with waking life that makes 
us condemn them. Certain uniformities are observed 
in waking life, while dreams seem quite erratic. The 
natural hypothesis would be that demons and the spirits 
of the dead visit us while we sleep ; but the modern 
mind, as a rule, refuses to entertain this view, though 
it is hard to see what could be said against it. On the 
other hand, the mystic, in moments of illumination, 
seems to awaken from a sleep which has filled all his 
mundane life : the whole world of sense becomes 
phantasmal, and he sees, with the clarity and convinc- 
ingness that belongs to our morning realization after 
dreams, a world utterly different from that of our daily 
cares and troubles. Who shall condemn him ? Who 
shall justify him ? Or who shall justify the seeming 
solidity of the common objects among which we sup- 
pose ourselves to live ? 

The hypothesis that other people have minds must, 
I think, be allowed to be not susceptible of any very 
strong support from the analogical argument. At the 
same time, it is a hypothesis which systematizes 
a vast body of facts and never leads to any consequences 
which there is reason to think false. There is therefore 
nothing to be said against its truth, and good reason to 
use it as a working hypothesis. When once it is 
admitted, it enables us to extend our knowledge of 
the sensible world by testimony, and thus leads to the 


system of private worlds which we assumed in our 
hypothetical construction. In actual fact, whatever 
we may try to think as philosophers, we cannot help 
believing in the minds of other people, so that the 
question whether our belief is justified has a merely 
speculative interest. And if it is justified, then there is 
no further difficulty of principle in that vast extension 
of our knowledge, beyond our own private data, which 
we find in science and common sense. 

This somewhat meagre conclusion must not be 
regarded as the whole outcome of our long discussion. 
The problem of the connection of sense with objective 
reality has commonly been dealt with from a standpoint 
which did not carry initial doubt so far as we have 
carried it ; most writers, consciously or unconsciously, 
have assumed that the testimony of others is to be 
admitted, and therefore (at least by implication) that 
others have minds. Their difficulties have arisen 
after this admission, from the differences in the appear- 
ance which one physical object presents to two people 
at the same time, or to one person at two times between 
which it cannot be supposed to have changed. Such 
difficulties have made people doubtful how far objective 
reality could be known by sense at all, and have made 
them suppose that there were positive arguments 
against the view that it can be so known. Our hypo- 
thetical construction meets these arguments, and 
shows that the account of the world given by common 
sense and physical science can be interpreted in a way 
which is logically unobjectionable, and finds a place 
for all the data, both hard and soft. It is this hypotheti- 
cal construction, with its reconciliation of psychology 
and physics, which is the chief outcome of our 
discussion. Probably the construction is only in 
part necessary as an initial, assumption, and can be 


obtained from more slender materials by the logical 
methods of which we shall have an example in the 
definitions of points, instants, and particles ; but I 
do not yet know to what lengths this diminution in 
our initial assumptions can be carried. 




AMONG the objections to the reality of objects of sense, 
there is one which is derived from the apparent differ- 
ence between matter as it appears in physics and 
things as they appear in sensation. Men of science, 
for the most part, are willing to condemn immediate 
data as " merely subjective," while yet maintaining 
the truth of the physics inferred from those data. 
But such an attitude, though it may be capable of 
justification, obviously stands in need of it ; and the 
only justification possible must be one which exhibits 
matter as a logical construction from sense-data 
unless, indeed, there were some wholly a priori prin- 
ciple by which unknown entities could be inferred 
from such as are known. It is therefore necessary to 
find some way of bridging the gulf between the world 
of physics and the world of sense, and it is this problem 
which will occupy us in the present lecture. Physicists 
appear to be unconscious of the gulf, while psycholo- 
gists, who are conscious of it, have not the mathe- 
matical knowledge required for spanning it. The 
problem is difficult, and I do not know its solution in 
detail. All that I can hope to do is to make the 
problem felt, and to indicate the kind of methods by 
which a solution is to be sought. 


Let us begin by a brief description of the two con- 
trasted worlds. We will take first the world of physics, 
for, though the other world is given while the physical 
world is inferred, to us now the world of physics is 
the more familiar, the world of pure sense having 
become strange and difficult to rediscover. Physics 
started from the common-sense belief in fairly per- 
manent and fairly- rigid bodiestables and chairs, 
stones, mountains, the earth and moon and sun. 
This common-sense belief, it should be noticed, is a 
piece of audacious metaphysical theorizing; objects 
are not continually present to sensation, and it may 
be doubted whether they are there when they are not 
seen or felt. This problem, which has been acute 
since the time of Berkeley, is ignored by common 
sense, and has therefore hitherto been ignored by 
physicists. We have thus here a first departure from 
the immediate data of sensation, though it is a depar- 
ture merely by way of extension, and was probably 
made by our savage ancestors in some very remote 
prehistoric epoch. 

But tables and chairs, stones and mountains, are 
not quite permanent or quite rigid. Tables and chairs 
lose their legs, stones axe split by frost, and mountains 
are cleft by earthquakes and eruptions. Then there 
are other things, which seem material, and yet present 
almost no permanence or rigidity. Breath, smoke, 
clouds, are examples of such things so, in a lesser 
degree, are ice and snow ; and rivers and seas, though 
fairly permanent, are not in any degree rigid. Breath, 
smoke, clouds, and generally things that can be seen 
but not touched, were thought to be hardly real ; to 
this day the usual mark of a ghost is that it can be 
seen but not touched. Such objects were peculiar in 
the fact that they seemed to disappear completely, 


not merely to be transformed into something else. 
Ice and snow, when they disappear, are replaced by 
water ; and it required no great theoretical effort to 
invent the hypothesis that the water was the same 
thing as the ice and snow, but in a new form. Solid 
bodies, when they break, break into parts which are 
practically the same in shape and size as they were 
before. A stone can be hammered into a powder, 
but the powder consists of grains which retain the 
character they had before the pounding. Thus the 
ideal of absolutely rigid and absolutely permanent 
bodies, which early physicists pursued throughout 
the changing appearances, seemed attainable by 
supposing ordinary bodies to be composed of a vast 
number of tiny atoms. This billiard-ball view of 
matter dominated the imagination of physicists until 
quite modern times, until, in fact, it was replaced by 
the electromagnetic theory, which in its turn has 
developed into a new atomism. Apart from the special 
form of the atomic theory which was invented for the 
needs of chemistry, some kind of atomism dominated 
the whole of traditional dynamics, and was implied in 
every statement of its laws and axioms. 

The modern form of atomism regards all matter 
as composed of two kinds of units, electrons and protons, 
both indestructible. All electrons, so far as we can dis- 
cover, are exactly alike, and so are all protons. In 
addition to this form of atomicity, which is not very 
different from that of the Greeks except in being based 
upon experimental evidence, there is a wholly new 
form, introduced by the theory of quanta. Here the 
indivisible unit is a unit of " action," i.e. energy multi- 
plied by time, or mass multiplied by length multiplied 
by velocity. This is not at all the sort of quantity in 
which traditional notions had led us to expect atom- 


icity. But relativity makes this kind of atomicity 
less surprising, although so fax it cannot deduce any 
form of atomicity, either old or new, from its funda- 
mental axioms. Relativity has introduced a wholly 
novel analysis of physical concepts, and has made it 
easier than it formerly was to build a bridge from 
physics to sense-data. To make this dear, it will 
be necessary to say something about relativity. But 
before doing so, let us examine our problem from the 
other end, namely that of sense-data. 

In the world of immediate data nothing is per- 
manent ; even the things that we regard as fairly 
permanent, such as mountains, only become data 
when we see them, and are not immediately given as 
existing at other moments. So far from one all- 
embracing space being given, there are several spaces 
for each person, according to the different senses which 
may be called spatial. Experience teaches us to 
obtain one space from these by correlation, and 
experience, together with instinctive theorizing, teaches 
us to correlate our spaces with those which we believe 
to exist in the sensible world of other people. The 
construction of a single time offers less difficulty so 
long as we confine ourselves to one person's private 
world, but the correlation of one private time with 
another is a matter of great difficulty. While engaged 
in the necessary logical constructions, we can console 
ourselves with the knowledge that permanent things, 
space, and time have ceased to be, for relativity 
physics, part of the bare bones of the world, and are 
now admitted to be constructions. In attempting to 
construct them from sense-data and particulars struc- 
turally analogous to sense-data, we are, therefore, 
only pushing the procedure of relativity theory one 
stage further back. 


The belief in indestructible " things " very early 
took the form of atomism. The underlying motive in 
atomism was not, I think, any empirical success in 
interpreting phenomena, but rather an instinctive 
belief that beneath all the changes of the sensible 
world there must be something permanent and un- 
changing. This belief was, no doubt, fostered and 
nourished by its practical successes, culminating in 
the conservation of mass ; but it was not produced 
by these successes. On the contrary, they were 
produced by it. Philosophical writers on physics 
sometimes speak as though the conservation of some- 
thing or other were essential to the possibility of 
science, but this, I believe, is an entirely erroneous 
opinion. If the a priori belief in permanence had 
not existed, the same laws which are now formu- 
lated in terms of this belief might just as well 
have been formulated without it. Why should we 
suppose that, when ice melts, the water which replaces 
it is the same thing in a new form ? Merely because 
this supposition enables us to state the phenomena 
in a way which is consonant with our prejudices. 
What we really know is that, under certain conditions 
of temperature, the appearance we call ice is replaced 
by the appearance we call water. We can give laws 
according to which the one appearance will be succeeded 
by the other, but there is no reason except prejudice 
for regarding both as appearances of the same 

One task, if what has just been said is correct, 
which, confronts us in trying to connect the world of 
sense with the world of physics, is the task of recon- 
structing the conception of matter without the a 
priori beliefs which historically gave rise to it. In 
spite of the revolutionary results of modern physics, 


the empirical successes of the conception of matter 
show that there must be some legitimate conception 
which fulfils roughly the same functions. The time 
has hardly come when we can state precisely what 
this legitimate conception is, but we can see in a 
general way what it must be like. For this purpose, 
it is only necessary to take our ordinary common-sense 
statements and reword them without the assumption 
of permanent substance. We say, for example, that 
things change gradually sometimes very quickly, 
but not without passing through a continuous series 
of intermediate states, or at least an approximately 
continuous series, if the discontinuities of the quantum 
theory should prove -ultimate. What this means is 
that, given any sensible appearance, there will usually 
be, if we watch, a continuous series of appearances 
connected with the given one, leading on by imper- 
ceptible gradations to the new appearances which 
common sense regards as those of the same thing. 
Thus a thing may be defined as a certain series of 
appearances, connected with each other by continuity 
and by certain causal laws. In the case of slowly 
changing things, this is easily seen. Consider, say, a 
wall-paper which fades in the course of years. It is 
an effort not to conceive of it as one " thing " whose 
colour is slightly different at one time from what it is 
at another. But what do we really know about it ? 
We know that under suitable circumstances^-i.e. when 
we are, as is said, " in the room "we perceive certain 
colours in a certain pattern : not always precisely the 
same colours, but sufficiently similar to fed familiar. 
If we can state the laws according to which the colour 
varies, we can state all that is empirically verifiable ; 
the assumption that there is a constant entity, the 
wall-paper, which "has" these various colours at 


various times, is a piece of gratuitous metaphysics. 
We may, if we like, define the wall-paper as the series 
of its aspects. These are collected together by the 
same motives which led us to regard the wall-paper 
as one thing, namely a combination of sensible con- 
tinuity and causal connection. More generally, a 
" thing " will be defined as a certain series of aspects, 
namely those which would commonly be said to be 
o/the thing. To say that a certain aspect is an aspect 
of a certain thing will merely mean that it is one of 
those which, taken serially, are the thing. Everything 
will then proceed as before : whatever was verifiable 
is unchanged, but our language is so interpreted as to 
avoid an unnecessary metaphysical assumption of 

The above extrusion of permanent things affords an 
example of the maxim which inspires all scientific 
philosophizing, namely " Occam's razor " : Entities are 
not to be multiplied without necessity. In other 
words, in dealing with any subject-matter, find out 
what entities are undeniably involved, and state 
everything in terms of these entities. Very often the 
resulting statement is more complicated and difficult 
than one which, like common sense and most philo- 
sophy, assumes hypothetical entities whose existence 
there is no good reason to believe in. We find it easier 
to imagine a wall-paper with changing colours than to 
think merely of the series of colours ; but it is a mistake 
to suppose that what is easy and natural in thought is 
what is most free from unwarrantable assumptions, as 
the case of " things " very aptly illustrates. 

The above summary account of the genesis of 
"things," though it may be correct in outline, has 
omitted some serious difficulties which it is necessary 
briefly to consider. Starting from a world of helter- 


skelter sense-data, we wish to collect them into series, 
each of which can be regarded as consisting of the 
successive appearances of one " thing." There is, to 
begin with, some conflict between what common sense 
regards as one thing, and what physics regards an 
unchanging collection of particles. To common sense, 
a human body is one thing, but to science the matter 
composing it is continually changing. This conflict, 
however, is not very serious, and may, for our rough 
preliminary purpose, be largely ignored. The problem 
is : by what principles shall we select certain data 
from the chaos, and call them all appearances of the 
same thing ? 

A rough and approximate answer to this question 
is not very difficult. There are certain fairly stable 
collections of appearances, such as landscapes, the 
furniture of rooms, the faces of acquaintances. In 
these cases, we have little hesitation in regarding them 
on successive occasions as appearances of one thing or 
collection of things. But, as the Comedy of Errors 
illustrates, we may be led astray if we judge by mere 
resemblance. This shows that something more is 
involved, for two difierent things may have any degree 
of likeness up to exact similarity. 

Another insufficient criterion of one thing is con- 
tinuity. As we have already seen, if we watch what 
we regard as one changing thing, we usually find its 
changes to be continuous so fax as our senses can 
perceive. We are thus led to assume that, if we see 
two finitely different appearances at two different times, 
and if we have reason to regard them as belonging 
to the same thing, then there was a continuous series 
of intermediate states of that thing during the time 
when we were not observing it. And so it comes to be 
thought that continuity of change is necessary and 



sufficient to constitute one thing. But in fact it is 
neither. It is not necessary, because the unobserved 
states, in the case where our attention has not been 
concentrated on the thing throughout, are purely 
hypothetical, and cannot possibly be our ground for 
supposing the earlier and later appearances to belong 
to the same thing ; on the contrary, it is because we 
suppose this that we assume intermediate unobserved 
states. Continuity is also not sufficient, since we can, 
for example, pass by sensibly continuous gradations 
from any one drop of the sea to any other drop. The 
utmost we can say is that discontinuity during un- 
interrupted observation is as a rule a mark of difference 
between things, though even this cannot be said in 
such cases as sudden explosions. (We are speaking 
throughout of the immediate sensible appearance, 
counting as continuous whatever seems continuous, 
and as discontinuous whatever seems discontinuous.) 
The assumption of continuity is, however, success- 
fully made in physics. This proves something, though 
not anything of very obvious utility to our present 
problem : it proves that nothing in the known world 
(apart, possibly, from quantum phenomena) is incon- 
sistent with the hypothesis that all changes are really 
continuous, though from too great rapidity or from 
our lack of observation they may not always appear 
continuous. In this hypothetical sense, continuity or 
change which, though sudden, is in accordance with 
quantum principles, may be allowed to be a necessary 
conidtion if two appearances are to be classed as 
appearances of the same thing. But it is not a sufficient 
condition, as appears from the instances of the drops 
in the sea. Thus something more must be sought 
before we can give even the roughest definition of a 
" thing." 


What is wanted further seems to be something in 
the nature of fulfilment of causal laws. This statement 
as it stands, is very vague, but we will endeavour to 
give it precision. When I speak of " causal laws," I 
mean any laws which connect events at different times, 
or even, as a limiting case, events at the same time 
provided the connection is not logically demonstrable. 
In this very general sense, the laws of dynamics are 
causal laws, and so are the laws correlating the simul- 
taneous appearances of one " thing " to different 
senses. The question is : How do such laws help in 
the definition of a " thing " ? 

To answer this question, we must consider what it 
is that is proved by the empirical success of physics. 
What is proved is that its hypotheses, though un- 
verifiable where they go beyond sense-data, are at no 
point in contradiction with sense-data, but, on the 
contrary, are ideally such as to render all sense-data 
calculable from a sufficient collection of data all belong- 
ing to a given period of time. Now physics has found 
it empirically possible to collect sense-data into series, 
each series being regarded as belonging to one " thing," 
and behaving, with regard to the laws of physics, in 
a way in which series not belonging to one thing would 
in general not behave. If it is to be unambiguous 
whether two appearances belong to the same thing or 
not, there must be only one way of grouping appear- 
ances so that the resulting things obey the laws of 
physics. It would be very difficult to prove that this 
is the case, but for our present purposes we may let 
this point pass, and assume that there is only one 
way. We must include in our definition of a " thing " 
those of its aspects, if any, which are not observed. 
Thus we may lay down the following definition : 
Things are those series of aspects which obey the laws of 


physics. That such series exist is an empirical fact, 
which constitutes the verifiability of physics. 

It may still be objected that the "matter" of 
physics is something other than series of sense-data. 
Sense-data, it may be said, belong to psychology and 
are, at any rate in some sense, subjective, whereas 
physics is quite independent of psychological con- 
siderations, and does not assume that its matter only 
exists when it is perceived. 

To this objection there are two answers, both of 
some importance. 

(a) We have been considering, in the above account, 
the question of the verifiability of physics. Now 
verifiability is by no means the same thing as truth ; 
it is, in fact, something far more subjective and 
psychological. For a proposition to be verifiable, it 
is not enough that it should be true, but it must also 
be such as we can discover to be true. Thus verifiability 
depends upon our capacity for acquiring knowledge, 
and not only upon the objective truth. In physics, 
as ordinarily set forth, there is much that is unverifi- 
able : there are hypotheses as to (a) how things would 
appear to a spectator in a place where, as it happens, 
there is no spectator ; (j3) how things would appear 
at times when, in fact, they are not appearing to 
anyone ; (y) things which never appear at all. All 
these are introduced to simplify the statement of 
the causal laws, but none of them form an integral 
part of what is known to be true in physics. This 
brings us to our second answer. 

(b) If physics is to consist wholly of propositions 
known to be true, or at least capable of being proved 
or disproved, the three kinds of hypothetical entities 
we have just enumerated must all be capable of being 
exhibited as logical functions of sense-data. In order 


to show how this might possibly be done, let us recall 
the hypothetical Ldbnizian universe of Lecture III. 
In that universe, we had a number of perspectives, 
two of which never had any entity in common, but 
often contained entities which could be sufficiently 
correlated to be regarded as belonging to the same 
thing. We will call one of these an " actual " private 
world when there is an actual spectator to which it 
appears, and " ideal " when it is merely constructed 
on principles of continuity. A physical thing consists, 
at each instant, of the whole set of its aspects at that 
instant, in all the different worlds ; thus a momentary 
state of a thing is a whole set of aspects. An " ideal " 
appearance will be an aspect merely calculated, but 
not actually perceived by any spectator. An " ideal " 
state of a thing will be a state at a moment when all 
its appearances are ideal. An ideal thing will be one 
whose states at all times are ideal. Ideal appearances, 
states, and things, since they are calculated, must be 
functions of actual appearances, states, and things ; 
in fact, ultimately, they must be functions of actual 
appearances. Thus it is unnecessary, for the enuncia- 
tion of the laws of physics, to assign any reality to 
ideal elements : it is enough to accept them as logical 
constructions, provided we have means of knowing 
how to determine when they become actual. This, 
in fact, we have with some degree of approximation ; 
the starry heaven, for instance, becomes actual when- 
ever we choose to look at it. It is open to us to believe 
that the ideal dements exist, and there can be no 
reason for disbelieving this ; but unless in virtue of 
some a priori law we cannot know it, for empirical 
knowledge is confined to what we actually observe. 

We come now to the conception of space. Here it 
is of the greatest importance to distinguish sharply 


between the space of physics and the space of one 
man's experience. It is the latter that must concern 
us first. 

People who have never read any psychology seldom 
realize how much mental labour has gone into the 
construction of the one all-embracing space into which 
all sensible objects are supposed to fit. Kant, who 
was unusually ignorant of psychology, described space 
as " an infinite given whole," whereas a moment's 
psychological reflection shows that a space which is 
infinite is not given, while a space which can be called 
given is not infinite. What the nature of " given " 
space really is, is a difficult question, upon which 
psychologists are by no means agreed. But some 
general remarks may be made, which will suffice to 
show the problems, without taking sides on any 
psychological issue still in debate. 

The first thing to notice is that different senses have 
different spaces. The space of sight is quite different 
from the space of touch : it is only by experience in 
infancy that we learn to correlate them. In later life, 
when we see an object within reach, we know how to 
touch it, and more or less what it will fed like ; if we 
touch an object with our eyes shut, we know where we 
should have to look for it, and more or less what it 
would look like. But this knowledge is derived from 
early experience of the correlation of certain kinds of 
touch-sensations with certain kinds of sight-sensations. 
The one space into which both kinds of sensations fit 
is an intellectual construction, not a datum. And 
besides touch and sight, there are other kinds of 
sensation which give other, though less important 
spaces : these also have to be fitted into the one space 
by means of experienced correlations. And as in the 
case of things, so here : the one all-embracing space, 


though convenient as a way of speaking, need not be 
supposed really to exist. All that experience makes 
certain is the several spaces of the several senses 
correlated by empirically discovered laws. . The one 
space may turn out to be valid as a logical construction, 
compounded of the several spaces, but there is no good 
reason to assume its independent metaphysical reality. 

Another respect in which the spaces of immediate 
experience differ from the space of geometry and 
physics is in regard to points, The space of geometry 
and physics consists of an infinite number of points, but 
no one has ever seen or touched a point. If there are 
points in a sensible space, they must be an inference. 
It is not easy to see any way in which, as independent 
entities, they could be validly inferred from the data ; 
thus here again, we shall have, if possible, to find 
some logical construction, some complex assemblage 
of immediately given objects, which will have the 
geometrical properties required of points. It is cus- 
tomary to think of points as simple and infinitely small, 
but geometry in no way demands that we should flwilr 
of them in this way. All that is necessary for geometry 
is that they should have mutual relations possessing 
certain enumerated abstract properties, and it may be 
that an assemblage of data of sensation will serve this 
purpose. Exactly how this is to be done I do not 
yet know, but it seems fairly certain that it can be 

An illustrative method, simplified so as to be easily 
manipulated, has been invented by Dr. Whitehead 
for the purpose of showing how points might be manu- 
factured from sense-data together with other structur- 
ally analogous particulars. This method is set forth 
in his Principles of Natural Knowledge (Cambridge, 
1919) and Concept of Nature (Cambridge, 1920). It 


is impossible to explain this method more concisely 
than in those books, to which the reader is therefore 
referred. But a few words may be said by way of 
explaining the general principles underlying the method. 
We have first of all to observe that there are no infini- 
tesimal sense-data : any surface we can see, for example, 
must be of some finite extent. We assume that this 
applies, not only to sense-data, but to the whole of 
the stuff composing the world : whatever is not an 
abstraction has some finite spatio-temporal size, 
though we cannot discover a lower limit to the sizes 
that are possible. But what appears as one undivided 
whole is often found, under the influence of attention, 
to split up into parts contained within the whole. 
Thus one spatial datum may be contained within 
another, and entirely enclosed by the other. This 
relation of enclosure, by the help of some very natural 
hypotheses, will enable us to define a " point " as a 
certain set of spatial objects ; roughly speaking, the 
set will consist of aJl volumes which would naturally 
be said to contain the point. 

It should be observed that Dr. Whitehead's abstract 
logical methods are applicable equally to psychological 
space, physical space, time, and space-time. But 
as applied to psychological space, they do not yield 
continuity unless we assume that sense-data always 
contain parts which are not sense-data. Sense-data 
have a minimum size, below which nothing is experi- 
enced ; but Dr. Whitehead's methods postulate that 
there shall be no such minimum. We cannot therefore 
construct a continuum without assuming the existence 
of particulars which are not experienced. This, 
however, does not constitute a real difficulty, since 
there is no reason to suppose that the space of our 
immediate experience possesses mathematical con- 


tinuity. The full employment of Dr. Whitehead's 
methods, therefore, belongs rather to physical space 
than to the space of experience. This question will 
concern us again later, when we come to consider 
physical space-time and its partial correlation with the 
space and time of experience. 

A very interesting attempt to show the kinds of 
geometry that can be constructed out of the actual 
materials supplied in sensation will be found in Jean 
Nicod's La gfom&rie dans le monde sensible (Paris, 


The question of time, so long as we confine ourselves 
to one private world, is rather less complicated than 
that of space, and we can see pretty dearly how it 
might be dealt with by such methods as we have been 
considering. Events of which we are conscious do not 
last merely for a mathematical instant, but always for 
some finite time, however short. Even if there be a 
physical world such as the mathematical theory of 
motion supposes, impressions on our sense-organs 
produce sensations which are not merely and strictly 
instantaneous, and therefore the objects of sense of 
which we are immediately conscious are not strictly 
instantaneous. Instants, therefore, are not among 
the data of experience, and, if legitimate, must be 
either inferred or constructed. It is difficult to see 
how they can be validly inferred ; thus we are left 
with the alternative that they must be constructed. 
How is this to be done ? 

Immediate experience provides us with two time- 
relations among events : they may be simultaneous, 
or one may be earlier and the other later. These two 
are both part of the crude data ; it is not the case that 
only the events are given, and their time-order is added 
by our subjective activity. The time-order, within 


certain limits, is as much given as the events. In any 
story of adventure you will find such passages as the 
following : " With a cynical smile he pointed the 
revolver at the breast of the dauntless youth. ' At the 
word three I shall fire,' he said. The words one and 
two had already been spoken with a cool and deliberate 
distinctness. The word three was forming on his 
lips, M this moment a blinding flash of lightning 
rent the air." Here we have simultaneity not due, 
as Kant would have us believe, to the subjective 
mental apparatus of the dauntless youth, but given as 
objectively as the revolver and the lightning. And 
it is equally given in immediate experience that the 
words one and two come earlier than the flash. These 
time-relations hold between events which are not 
strictly instantaneous. Thus one event may begin 
sooner than another, and therefore be before it, but 
may continue after the other has begun, and therefore 
be also simultaneous with it. If it persists after the 
other is over, it will also be later than the other. 
Earlier, simultaneous, and later, are not inconsistent 
with each other when we are concerned with events 
which last for a finite time, however short ; they 
only become inconsistent when we are dealing with 
something instantaneous. 

It is to be observed that we cannot give what may 
be called absolute dates, but only dates determined by 
events. We cannot point to a time itseUE, but only 
to some event occurring at that time. There is 
therefore no reason in experience to suppose that there 
are times as opposed to events : the events, ordered 
by the relations of simultaneity and succession, axe 
all that experience provides. Hence, unless we are 
to introduce superfluous metaphysical entities, we 
must, in defining what we can regard as an instant, 


proceed by means of some construction which assumes 
nothing beyond events and their temporal relations. 

If we wish to assign a date exactly by means of events, 
how shall we proceed ? If we take any one event, we 
cannot assign our date exactly, because the event is 
not instantaneous, that is to say, it may be simultane- 
ous with two events which are not simultaneous with 
each other. In order to assign a date exactly, we must 
be able, theoretically, to determine whether any given 
event is before, at, or after this date, and we must 
know that any other date is either before or after this 
date, but not simultaneous with it. Suppose, now, 
instead of taking one event A, we take two events A 
and B, and suppose A and B partly overlap, but B 
ends before A ends. Then an event which is simul- 
taneous with both A and B must exist during the time 
when A and B overlap ; thus we have come rather 
nearer to a precise date than when we considered 
A and B alone. Let C be an event which is simul- 
taneous with both A and B, but which ends before 
either A or B has ended. Then an event which is 
simultaneous with A and B and C must exist during 

the time when all three overlap, which is a still 
shorter time. Proceeding in this way, by taking more 
and more events, a new event which is dated as 
simultaneous with all of them becomes gradually 
more and more accurately dated. This suggests a 


way by which a completely accurate date can be 

Let us take a group of events of which any two 
overlap, so that there is some time, however short, 
when they all exist. If there is any other event which 
is simultaneous with all of these, let us add it to the 
group ; let us go on until we have constructed a 
group such that no event outside the group is simul- 
taneous with all of them, but all the events inside the 
group are simultaneous with each other. Let us 
define this whole group as an instant of time. It 
remains to show that it has the properties we expect 
of an instant. 

What are the properties we expect of instants ? 
First, they must form a series : of any two, one must 
be before the other, and the other must be not before 
the one ; if one is before another, and the other before 
a third, the first must be before the third. Secondly, 
every event must be at a certain number of instants ; 
two events are simultaneous if they are at the same 
instant, and one is before the other if there is an instant, 
at which the one is, which is earlier than some instant 
at which the other is. Thirdly, if we assume that 
there is always some change going on somewhere 
during the time when any given event persists, the 
series of instants ought to be compact, i.e. given any 
two instants, there ought to be other instants between 
them. Do instants, as we have defined them, have 
these properties ? 

We shall say that an event is " at " an instant when 
it is a member of the group by which the instant is 
constituted; and we shall say that one instant is 
before another if the group which is the one instant 
contains an event which is earlier than, but not simul- 
taneous with, some event in the group which is the 


other instant. When one event is earlier than, but 
not simultaneous with another, we shall say that it 
" wholly precedes " the other. Now we know that 
of two events which belong to one experience but are 
not simultaneous, there must be one which wholly 
precedes the other, and in that case the other cannot 
also wholly precede the one ; we also know that, if 
one event wholly precedes another, and the other 
wholly precedes a third, then the first wholly pre- 
cedes the third. From these facts it is easy to deduce 
that the instants as we have defined them form 
a series. 

We have next to show that every event is " at " 
least one instant, i.e. that, given any event, there is 
at least one class, such as we used in defining instants, 
of which it is a member. For this purpose, consider 
all the events which are simultaneous with a given 
event, and do not begin later, Le. are not wholly 
after anything simultaneous with it. We will call 
these the " initial contemporaries of the given event. 
It will be found that this class of events is the first 
instant at which the given event exists, provided 
every event wholly after some contemporary of the 
given event is wholly after some initial contemporary 
of it. 

Finally, the series of instants will be compact if, 
given any two events of which one wholly precedes 
the other, there are events wholly after the one and 
simultaneous with something wholly before the other. 
Whether this is the case or not, is an empirical question ; 
but if it is not, there is no reason to expect the time- 
series to be compact. 1 

1 The assumptions made concerning time-relations in one 
experience in the above axe as follows : 

I. In order to secure that instants form a series, we assume : 


way by which a completely accurate date can be 

Let us take a group of events of which any two 
overlap, so that there is some time, however short, 
when they all exist. If there is any other event which 
is simultaneous with all of these, let us add it to the 
group ; let us go on until we have constructed a 
group such that no event outside the group is simul- 
taneous with all of them, but all the events inside the 
group are simultaneous with each other. Let us 
define this whole group as an instant of time. It 
remains to show that it has the properties we expect 
of an instant. 

What are the properties we expect of instants? 
First, they must form a series : of any two, one must 
be before the other, and the other must be not before 
the one ; if one is before another, and the other before 
a third, the first must be before the third. Secondly, 
every event must be at a certain number of instants ; 
two events are simultaneous if they are at the same 
instant, and one is before the other if there is an instant, 
at which the one is, which is earlier than some instant 
at which the other is. Thirdly, if we assume that 
there is always some change going on somewhere 
during the time when any given event persists, the 
series of instants ought to be compact, i.e. given any 
two instants, there ought to be other instants between 
them. Do instants, as we have defined them, have 
these properties ? 

We shall say that an event is " at " an instant when 
it is a member of the group by which the instant is 
constituted; and we shall say that one instant is 
before another if the group which is the one instant 
contains an event which is earlier than, but not simul- 
taneous with, some event in the group which is the 


other instant. When one event is earlier than, but 
not simultaneous with another, we shall say that it 
" wholly precedes " the other. Now we know that 
of two events which belong to one experience but are 
not simultaneous, there must be one which wholly 
precedes the other, and in that case the other cannot 
also wholly precede the one ; we also know that, if 
one event wholly precedes another, and the other 
wholly precedes a third, then the first wholly pre- 
cedes the third. From these facts it is easy to deduce 
that the instants as we have defined them form 
a series. 

We have next to show that every event is "at" 
least one instant, i.e. that, given any event, there is 
at least one class, such as we used in defining instants, 
of which it is a member. For this purpose, consider 
all the events which are simultaneous with a given 
event, and do not begin later, i.e. are not wholly 
after anything simultaneous with it. We will call 
these the " imtia.l contemporaries of the given event. 
It will be found that this class of events is the first 
instant at which the given event exists, provided 
every event wholly after some contemporary of the 
given event is wholly after some initial contemporary 
of it. 

Finally, the series of instants will be compact if, 
given any two events of which one wholly precedes 
the other, there are events wholly after the one and 
simultaneous with something wholly before the other. 
Whether this is the case or not, is an empirical question ; 
but if it is not, there is no reason to expect the time- 
series to be compact. 1 

i The assumptions made concerning tune-relations in one 
experience in the above are as follows : 

I. In older to secure that instants form a series, we assume : 


Thus our definition of instants secures all that 
mathematics requires, without having to assume the 
existence of any disputable metaphysical entities. 

With regard to compactness in the time of one 
experience, there are the same observations to make 
as in the case of space. The events which we experi- 
ence have not only a finite duration, but a duration 
which cannot sink below a certain TniniTnnni ; there- 
fore they will only fit into a compact series if we either 
bring in events wholly outside our experience, or 
assume that experienced events have parts which we 
do not experience, or postulate that we can experi- 

(*) No event wholly precedes itself. (An " event " is 
defined as whatever is simultaneous with some- 
thing or other.) 

(6) If one event wholly precedes another, and the other 
wholly precedes a third, then the first wholly 
precedes the third. 

(e) If one event wholly precedes another, it is not 
simultaneous with it. 

(d) Of two events which are not simultaneous, one 

must wholly precede the other. 

II. In order to secure that the initial contemporaries of a 
given event should form an instant, we assume : 

(e ) An event wholly after some contemporary of a given 

event is wholly after some initial contemporary 

of the given event. 
III. In order to secure that the series of instants shall be 

compact, we assume : 
(/) If one event wholly precedes another, there is an 

event wholly after the one and simultaneous with 

something wholly before the other. 

This assumption entails the consequence that if one event 
covers the whole of a stretch of time immediately preceding 
another event, then it must have at least one instant in common 
with the other event ; i.e. it is impossible for one event to cease 
just before another begins. I do not know whether this should 
be regarded as inadmissible. For a mathematico-logical 
treatment of the above topics, cf . K. Wiener, " A Contribution 
to the Theory of Relative Position," Proc. Camb. Phil. Soc., 
xvii. 5, pp. 441-449. 


ence an Infinite number of events at once. Here, 
again, the full application of our logico-mathematical 
method is only possible when we come to physical time. 
This topic will be discussed again near the end of 
Lecture V. 

Instants may also be defined by means of the 
enclosure-relation, exactly as was done in the case of 
points. One object will be temporally enclosed by 
another when it is simultaneous with the other, but 
not before or after it. Whatever encloses temporally 
or is enclosed temporally we shall call an " event." 
In order that the relation of temporal enclosure may 
lead to instants we require (i) that it should be tran- 
sitive, i.e. that if one event encloses another, and 
the other a third, then the first encloses the third ; 
(2) that every event encloses itself, but if one event 
encloses another different event, then the other does 
not enclose the one ; (3) that given any set of events 
such that there is at least one event enclosed by all 
of them, then there is an event enclosing all that they 
all enclose, and itself enclosed by all of them ; (4) that 
there is at least one event. To ensure infinite divisi- 
bility, we require also that every event should enclose 
events other than itself. Assuming these character- 
istics, temporal enclosure can be made to give rise to a 
compact series of instants. We can now form an " en- . 
closure-series " of events, by choosing a group of events 
such that of any two there is one which encloses the 
other ; this will be a " punctual enclosure-series " if, 
given any other enclosure-series such that every member 
of our first series encloses some member of our second, 
then every member of our second series encloses some 
member of our first. Then an " instant " is the class 
of all events which enclose members of a given punctual 


The correlation of the times of different private 
worlds is a more difficult matter. We saw, in Lecture 
III, that different private worlds often contain corre- 
lated appearances, such as common sense would 
regard as appearances of the same "thing." When 
two appearances in different worlds are so correlated 
as to belong to one momentary " state " of a thing, 
it would be natural to regard them as simultaneous, 
and as thus affording a simple means of correlating 
different private times. But this can only be regarded 
as a first approximation. What we call one sound 
will be heard sooner by people near the source of the 
sound than by people further from it, and the same 
applies, though in a less degree, to light. Thus two 
correlated appearances in different worlds are not 
necessarily to be regarded as occurring at the same 
date in physical time, though they will be parts of one 
momentary state of a thing. The correlation of 
different private times is regulated by the desire to 
secure the simplest possible statement of the laws of 
physics, and thus raises rather complicated technical 
problems; these problems are dealt with by the 
theory of relativity, and show that it is impossible 
validly to construct one all-embracing time having 
any physical significance. 

The above brief outline, must not be regarded as 
more than tentative and suggestive. It is intended 
merely to show the kind of way in which, given a 
world with the kind of properties that psychologists 
find in the world of sense, it may be possible, by 
means of purely logical constructions, to make it 
amenable to mathematical treatment by defining 
series or classes of sense-data which can be called 
respectively particles, points, and instants. If such 
constructions are possible, then mathematical physics 


is applicable to the real world, in spite of the fact that 
its particles, points, and instants are not to be found 
among actually existing entities. 

The space-time of physics has not a very dose 
relation to the space and time of the world of one 
person's experience. Everything that occurs in one 
person's experience must, from the standpoint of 
physics, be located within that person's body ; this 
is evident from considerations of causal continuity. 
What occurs when I see a star occurs as the result of 
light-waves impinging on the retina, and causing a 
process in the optic nerve and brain ; therefore the 
occurrence called "seeing a star" must be in the 
brain. If we define a piece of matter as a set of events 
(as was suggested above), the sensation of seeing a 
star will be one of the events which are the brain of 
the percipient at the time of the perception. Thus 
every event that I experience will be one of the events 
which constitute some part of my body. The space 
of (say) my visual perceptions is only correlated, with 
physical space, more or less approximately ; from the 
physical point of view, whatever I see is inside my 
head. I do not see physical objects; I see effects 
which they produce in the region where my brain is. 
The correlation of visual and physical space is rendered 
approximate by the fact that my visual sensations are 
not wholly due each to some physical object, but also 
partly to the intervening medium. Further, the rela- 
tion of visual sensation to physical object is one-many, 
not not-one, because our senses are more or less vague : 
things which look different under the microscope may 
be indistinguishable to the naked eye. The inferences 
from perceptions to physical facts depend always upon 
causal laws, which enable us to bring past history to 
bear ; e.g. if we have just examined an object under a 



microscope, we assume that it is still very similar to 
what we then saw it to be, or rather, to what we 
inferred it to be from what we then saw. It is through 
history and testimony, together with causal laws, 
that we arrive at physical knowledge which is much 
more precise than anything inferable from the percep- 
tions of one moment. History, testimony, and causal 
laws axe, of course, in their various degrees, open to 
question. But we are not now considering whether 
physics is true, but how, if it is true, its world is related 
to that of the senses. 

With regard to time, the relation of psychology to 
physics is surprisingly simple. The time of our 
experience is the time which results, in physics, from 
taking our own body as the origin. Seeing that 
all the events in my experience are, for physics, in 
my body, the time-interval between them is what 
relativity theory calls the " interval " (in space-time) 
between them. Thus the time-interval between two 
events in one person's experience retains a direct 
physical significance in the theory of relativity. But 
the merging of physical space and time into space- 
time does not correspond to anything in psychology. 
Two events which are simultaneous in my experience 
may be spatially separate in psychical space, e.g. 
when I see two stars at once. But in physical space 
these two events are not separated, and indeed they 
occur in the same place in space-time. Thus in this 
respect relativity theory has complicated the relation 
between perception and physics. 

The problem which the above considerations are 
intended to elucidate is one whose importance and 
even existence has been concealed by the unfortunate 
separation of different studies which prevails through- 
out the civilized world. Physicists, ignorant and con- 


temptuous of philosophy, have been content to assume 
.their particles, points, and instants in practice, while 
conceding, with ironical politeness, that their concepts 
laid no claim to metaphysical validity. Metaphysi- 
cians, obsessed by the idealistic opinion that only mind 
is real, and the Parmenidean belief that the real is 
unchanging, repeated one after another the supposed 
contradictions in the notions of matter, space, and 
time, and therefore naturally made no endeavour to 
invent a tenable theory of particles, points, and 
instants. Psychologists, who have done invaluable 
work in bringing to light the chaotic nature of the 
crude materials supplied by unmanipulated sensation, 
have been ignorant of mathematics and modern 
logic, and have therefoie been content to say that 
matter, space, and time are "intellectual construc- 
tions," without making any attempt to show in detail 
either how the intellect can construct them, or what 
secures the practical validity which physics shows 
them to possess. Philosophers, it is to be hoped, will 
come to recognize that they cannot achieve any solid 
success in such problems without some slight knowledge 
of logic, mathematics, and physics ; meanwhile, for 
want of students with the necessary equipment, this 
vital problem remains unattempted and unknown. 1 

There are, it is true, two authors, both physicists, 
whotave done something, though not much, to bring 
about a recognition of the problem as one demanding 
study. These two authors are Poincarfi and Mach, 
Poincar6 especially in his Science and Hypothesis, 

* This was written in 1914. Since then, largely as a result 
of the general theory of relativity, a great deal of valuable 
work has been done; I should wish to mention specially 
Professor Eddington, Dr. Whitehead, and Dr. Broad, as having 
contributed, from different angles, to the solution of the 
problems dealt with in this lecture. 


Mach especially in his Analysis of Sensations. Both 
of them, however, admirable as their work is, seem to 
me to suffer from a general philosophical bias. Poin- 
car6 is Kantian, while Mach is ultra-empiricist ; with 
Poincar6 almost all the mathematical part of physics 
is merely conventional, while with Mach the sensation 
as a mental event is identified with its object as a 
part of the physical world. Nevertheless, both these 
authors, and especially Mach, deserve mention as 
having made serious contributions to the consideration 
of our problem. 

When a point or an instant is defined as a class of 
sensible qualities, the first impression produced is 
likely to be one of wild and wilful paradox. Certain 
considerations apply here, however, which will again be 
relevant when we come to the definition of numbers. 
There is a whole type of problems which can be solved 
by such definitions, and almost always there will be 
at first an effect of paradox. Given a set of objects 
any two of which have a relation of the sort called 
"symmetrical and transitive," it is almost certain 
that we shall come to regard them as all having some 
common quality, or as all having the same relation 
to some one object outside the set. This kind of case 
is important, and I shall therefore try to make it 
clear even at the cost of some repetition of previous 

A relation is said to be " symmetrical " when, if one 
term has this relation to another, then the other also 
has it to the one. Thus "brother or sister" is a 
1 ' symmetrical " relation : if one person is a brother or 
a sister of another, then the other is a brother or 
sister of the one. Simultaneity, again, is a symmetrical 
relation ; so is equality in size. A relation is said to 
be " transitive " when, if one term has this relation to 


another, and the other to a third, then the one has it 
to the third. The symmetrical relations mentioned 
just now are also transitive provided, in the case of 
" brother or sister," we allow a person to be counted 
as his or her own brother or sister, and provided, in 
the case of simultaneity, we mean complete simul- 
taneity, i.e. beginning and ending together. 

But many relations are transitive without being 
symmetrical for instance, such relations as " greater," 
" earlier," "to the right of," " ancestor of," in fact 
all such relations as give rise to series. Other relations 
are symmetrical without being transitive for example, 
difference in any respect. If A is of a different age 
from B, and B of a different age from C, it does not 
f ollow that A is of a different age from C. Simultaneity, 
again, in the case of events which last for a finite time, 
will not necessarily be transitive if it only means that 
the times of the two events overlap. If A ends just 
after B has begun, and B ends just after C has begun, 
A and B will be simultaneous in this sense, and so will 
B and C, but A and C may well not be simultaneous* 

All the relations which can naturally be represented 
as equality in any respect, or as possession of a common 
property, are transitive and symmetrical this applies, 
for example, to such relations as being of the same 
height or weight or colour. Owing to the fact that 
possession of a common property gives rise to a transi- 
tive symmetrical relation, we come to imagine that 
wherever such a relation occurs it must be due to a 
common property. " Being equally numerous " is a 
transitive symmetrical relation of two collections; 
hence we imagine that both have a common property, 
called their number. " Existing at a given instant " 
(in the sense in which we defined an instant) is a 
transitive symmetrical relation; hence we come to 


think that there really is an instant which confers a 
common property on all the things existing at that 
instant. " Being states of a given thing " is a transi- 
tive symmetrical relation ; hence we come to imagine 
that there really is a thing, other than the series of 
states, which accounts for the transitive symmetrical 
relation. In all such cases, the class of terms that 
have the given transitive symmetrical relation to a 
given term will fulfil all the formal requisites of a 
common property of all the members of the class. 
Since there certainly is the class, while any other 
common property may be illusory, it is prudent, in 
order to avoid needless assumptions, to substitute the 
class for the common property which would be ordin- 
arily assumed. This is the reason for the definitions 
we have adopted, and this is the source of the apparent 
paradoxes. No harm is done if there are such common 
properties as language assumes, since we do not deny 
them, but merely abstain from asserting them. But 
if there are not such common properties in any given 
case, then our method has secured us against error. 
In the absence of special knowledge, therefore, the 
method we have adopted is the only one which is 
safe, and which avoids the risk of introducing fictitious 
metaphysical entities. 


THE theory of continuity, with which we shall be 
occupied in the present lecture, is, in most of its 
refinements and developments, a purely mathematical 
subject very beautiful, very important, and very 
delightful, but not, strictly speaMng, a part of philo- 
sophy. The logical basis of the theory alone belongs 
to philosophy, and alone will occupy us to-night. 
The way the problem of continuity enters into philo- 
sophy is, broadly speaking, the following : Space and 
time are treated by Mhgpiatiffo-Ti s as consisting of 
points and instants, but they also have a property, 
easier to feel than to define, which is called continuity, 
and is thought by many philosophers to be destroyed 
when they are resolved into points and instants. 
Zeno, as we shall see, proved that analysis into points 
and instants was impossible if we adhered to the 
view that the number of points or instants in a finite 
space or time must be finite. Later philosophers, 
believing infinite number to be self-contradictory, have 
found here an antinomy : Spaces and times could not 
consist of a finite number of points and instants, for 
such reasons as Zeno's ; they could not consist of 
an infinite number of points and instants, because 
infinite numbers were supposed to be self-contradictory. 
Therefore spaces and times, if real at all, must not be 
regarded as composed of points anci instants, 


But even when points and instants, as independent 
entities, are discarded, as they were by the theory 
advocated in our last lecture, the problems of con- 
tinuity, as I shall try to show presently, remain, in a 
practically unchanged form. Let us therefore, to 
begin with, admit points and instants, and consider 
the problems in connection with this simpler or at 
least more familiar hypothesis. 

The argument against continuity, in so far as it 
rests upon the supposed difficulties of infinite numbers, 
has been disposed of by the positive theory of the 
infinite, which will be considered in Lecture VII. 
But there remains a feeling of the kind that led 
Zeno to the contention that the arrow in its flight is 
at rest which suggests that points and instants, even 
if they are infinitely numerous, can only give a jerky 
motion, a succession of different immobilities, not 
the smooth transitions with which the senses have 
made us familiar. This feeling is due, I believe, to 
a failure to realize imaginatively, as well as abstractly, 
the nature of continuous series as they appear in 
mathematics. When a theory has been apprehended 
logically, there is often a long and serious labour still 
required in order to feel it : it is necessary to dwell 
upon it, to thrust out from the mind, one by one, the 
misleading suggestions of false but more familiar 
theories, to acquire the kind of intimacy which, in 
the case of a foreign language, would enable us to 
think and dream in it, not merely to construct laborious 
sentences by the help of grammar and dictionary. 
It is, I believe, the absence of this kind of intimacy 
which makes many philosophers regard the mathe- 
matical doctrine of continuity as an inadequate 
explanation of the continuity which we experience 
in the world of sense. 


In the present lecture, I shall first try to explain 
in outline what the mathematical theory of continuity 
is in its philosophically important essentials. The 
application to actual space and time will not be in 
question to begin with. I do not see any reason to 
suppose that the points and instants which mathe- 
maticians introduce in dealing with space and time are 
actual physically existing entities, but I do see reason 
to suppose that the continuity of actual space and 
time may be more or less analogous to mathematical 
continuity. The theory of mathematical continuity 
is an abstract logical theory, not dependent for its 
validity upon any properties of actual space and time. 
What is claimed for it is that, when it is understood, 
certain characteristics of space and time, previously 
very hard to analyse, are found not to present any 
logical difficulty. What we know empirically about 
space and time is insufficient to enable us to decide 
between various mathematically possible alternatives, 
but these alternatives are all fully intelligible and 
fully adequate to the observed facts. For the present, 
however, it will be well to forget space and time and 
the continuity of sensible change, in order to return 
to these topics equipped with the weapons provided 
by the abstract theory of continuity. 

Continuity, in mathematics, is a property only 
possible to a series of terms, i.e. to terms arranged in 
an order, so that we can say of any two that one comes 
before the other. Numbers in order of magnitude, the 
points on a line from left to right, the moments of 
time from earlier to later, are instances of series. The 
notion of order, which is here introduced, is one which 
is not required in the theory of cardinal number. 
It is possible to know that two classes have the same 
number of terms without knowing any order in which 


they are to be taken. We have an instance of this in 
such a case as English husbands and English wives : 
we can see that there must be the same number of 
husbands as of wives, without having to arrange them 
in a series. But continuity, which we are now to 
consider, is essentially a property of an order: it 
does not belong to a set of terms in themselves, but 
only to a set in a certain order. A set of terms which 
can be arranged in one order can always also be arranged 
in other orders, and a set of terms which can be arranged 
in a continuous order can always be arranged in orders 
which are not continuous. Thus the essence of con- 
tinuity must not be sought in the nature of the set 
of terms, but in the nature of their arrangement in a 

Mathematicians have distinguished different degrees 
of continuity, and have confined the word " con- 
tinuous," for technical purposes, to series having a 
certain high degree of continuity. But for philoso- 
phical purposes, all that is important in continuity is 
introduced by the lowest degree of continuity, which 
is called "compactness." A series is called 
"compact" when no two terms are consecutive, 
but between any two there are others. One of the 
simplest examples of a compact series is the series 
of fractions in order of magnitude. Given any two 
fractions, however near together, there are other 
fractions greater than the one and smaller than the 
other, and therefore no two fractions are consecutive. 
There is no fraction, for example, which is next after J : 
if we choose some fraction which is very little greater 
Ibs* i sa Y iVb* TO can find others, such as J-, which 
are nearer to . Thus between any two fractions, 
however little they differ, there are an infinite "number 
of other fractions. Mathematical space and tjm$ 


also have this property of compactness, though whether 
actual space and tune have it is a further question, 
dependent upon empirical evidence, and probably 
incapable of being answered with certainty. 

In the case of abstract objects such as fractions, it 
is perhaps not very difficult to realize the logical 
possibility of their forming a compact series. The 
difficulties that might be felt are those of infinity, for 
in a compact series the number of terms between 
any two given terms must be infinite. But when these 
difficulties have been solved, the mere compactness in . 
itself offers no great obstacle to the imagination. In 
more concrete cases, however, such as motion, com- 
pactness becomes much more repugnant to our habits 
of thought. It will therefore be desirable to consider 
explicitly the mathematical account of motion, with 
a view to making its logical possibility felt. The 
mathematical account of motion is perhaps artificially 
simplified when regarded as describing what actually 
occurs in the physical world; but what actually 
occurs must be capable, by a certain amount of logical 
manipulation, of being brought within the scope of 
the mathematical account, and must, in its analysis, 
raise just such problems as are raised in their simplest 
form by this account. Neglecting, therefore, for the 
present, the question of its physical adequacy, let 
us devote ourselves merely to considering its possibility 
as a formal statement of the nature of motion. 

In order to simplify our problem as much as possible, 
let us imagine a tiny speck of light moving along a 
scale. What do we mean by saying that the motion 
is continuous ? It is not necessary for our purposes 
to consider the whole of what the mathematician 
means by this statement : only part of what he means 
is philosophically important. One part of what he 


means is that, if we consider any two positions of 
the speck occupied at any two instants, there will be 
other intermediate positions occupied at intermediate 
instants. However near together we take the two 
positions, the speck will not jump suddenly from 
the one to the other, but will pass through an infinite 
number of other positions on the way. Every dis- 
tance, however small, is traversed by passing through 
all the infinite series of positions between the two ends 
of the distance. 

But at this point imagination suggests that we may 
describe the continuity of motion by saying that the 
speck always passes from one position at one instant 
to the next position at the next instant. As soon as 
we say this or imagine it, we faJl into error, because 
there is no next point or next instant. If there were, 
we should find Zeno's paradoxes, in some form, un? 
avoidable, as will appear in our next lecture. One 
simple, paradox may serve as an illustration. If our 
speck is in motion along the scale throughout the 
whole of a certain time, it cannot be at the same point 
at two consecutive instants. But it cannot, from one 
instant to the next, travel further than from one point 
to the next, for if it did, there would be no instant at 
which it was in the positions intermediate between 
that at the first instant and that at the next, and 
we agreed that the continuity of motion excludes the 
possibility of such sudden jumps. It follows that our 
speck must, so long as it moves, pass from one point 
at one instant to the next point at the next instant. 
Thus there will be just one perfectly definite velocity 
with which all motions must take place : no motion 
can be faster than this, and no motion can be slower. 
Since this conclusion is false, we must reject the hypo- 
thesis upon which it is based, namely that there are 


consecutive points and instants. 1 Hence the con- 
tinuity of motion must not be supposed to consist 
in a body's occupying consecutive positions at con- 
secutive times. 

The difficulty to imagination lies chiefly, I think, in 
keeping out the suggestion of infinitesimal distances 
and times. Suppose we halve a given distance, and 
then halve the half, and so on, we can continue the 
process as long as we please, and the longer we con- 
tinue it, the smaller the resulting distance becomes. 
This infinite divisibility seems, at first sight, to imply 
that there are infinitesimal distances, i.e. distances 
so small that any finite fraction of an inch would be 
greater. This, however, is an error. The continued 
bisection of our distance, though it gives us continually 
smaller distances, gives us always finite distances. If 
pur original distance was an inch, we reach successively 
half an inch, a quarter of an inch, an eighth, a six- 
teenth, and so on; but every one of this infinite 
series of diminishing distances is finite, " But/ 1 it 
may be said, " in the end the distance will grow infini- 
tesimal." No, because there is no end. The process 
of bisection is one which can, theoretically, be carried 
on for ever, without any last term being attained. 
Thus infinite divisibility of distances, which must be 
admitted, does not imply that there are distances so 
small that any finite distance would be larger. 

It is easy, in this kind of question, to fall into an 
elementary logical blunder. Given any finite dis- 
tance, we can find a smaller distance ; this may be 
expressed in the ambiguous fonn " there is a distance 
smaller than any finite distance." But if this is then 

i The above paradox is essentially the same as Zeno's 
argument of the stadium which will be considered in our next 


interpreted as meaning " there is a distance such that, 
whatever finite distance may be chosen, the distance 
in question is smaller," then the statement is false. 
Common language is ill adapted to expressing matters 
of this kind, and philosophers who have been dependent 
on it have frequently been misled by it. 

In a continuous motion, then, we shall say that at any 
given instant the moving body occupies a certain posi- 
tion, and at other instants it occupies other positions ; 
the interval between any two instants and between 
any two positions is always finite, but the continuity 
of the motion is shown in the fact that, however 'near 
togetherwe take the two positions and the two instants, 
there are an infinite number of positions still nearer 
together, which are occupied at instants that are also 
still nearer together. The moving body never jumps 
from one position to another, but always passes by 
a gradual transition through an infinite number of 
intermediaries. At a given instant, it is where it is, 
like Zeno's arrow ; * but we cannot say that it is at 
rest at the instant, since the instant does not last for 
a finite time, and there is not a beginning and end 
of the instant with an interval between them. Rest 
consists in being in the same position at all the instants 
throughout a certain finite period, however short ; it 
does not consist simply in a body's being where it is 
at a given instant. This whole theory, as is obvious, 
depends upon the nature of compact series, and 
demands, for its full comprehension, that compact 
series should have become familiar and easy to the 
imagination as well as to deliberate thought. 

What is required may be expressed in mathematical 
language by saying that the position of a moving body 
must be a continuous function of the time. To define 
See next lecture. 


accurately what this means, we proceed as follows. 
Consider a particle which, at the moment t, is at the 

- ' 

point P. Choose now any small portion PiP 2 
of the path of the particle, this portion being 
one which contains P. We say then that, if the 
motion of the particle is continuous at the time *, 
it must be possible to find two instants ^, t^, one 
earlier than t and one later, such that throughout 
the whole time from ^ to 2 (both included), the 
particle lies between P! and P 2 . And we say that 
this must still hold however small we make the portion 
P! P a . When this is the case, we say that the motion 
is continuous at the time t ; and when the motion is. 
continuous at all times, we say that the motion as a 
whole is continuous. It is obvious that if the particle 
were to jump suddenly from P to some other point 
Q, our definition would fail for all intervals PI PS 
which were too small to include Q. Thus our definition 
affords an analysis of the continuity of motion, while 
admitting points and instants and denying infinitesimal 
distances in space or periods in time. 

Philosophers, mostly in ignorance of the mathe- 
matician's analysis, have adopted other and more 
heroic methods of dealing with the prima fade diffi- 
culties of continuous motion. A typical and recent 
example of philosophic theories of motion is afforded 
by Bergson, whose views on this subject I have 
examined elsewhere. 1 

Apart from definite arguments, there are certain 
feelings, rather than reasons, which stand in the way 
of an acceptance of the mathematical account of 
* Monist, July 1912, pp. 337-341. 


motion. To begin with, if a body is moving at all 
fast, we see its motion just as we see its colour. A 
slow motion, like that of the hour-hand of a watch, 
is only known in the way which mathematics would 
lead us to expect, namely by observing a change of 
position after a lapse of time ; but, when we observe 
the motion of the second-hand, we do not merely 
see first one position and then another we see some- 
thing as directly sensible as colour. What is this 
something that we see, and that we call visible motion ? 
Whatever it is, it is not the successive occupation of 
successive positions : something beyond the mathe- 
matical theory of motion is required to account for it. 
Opponents of the mathematical theory emphasize this 
fact. "Your theory," they say, "may be very 
logical, and might apply admirably to some other 
world ; but in this actual world, actual motions are 
quite different from what your theory would declare 
them to be, and require, therefore, some different 
philosophy from yours for their adequate explanation." 

The objection thus raised is one which I have no 
wish to underrate, but I believe it can be fully answered 
without departing from the methods and the outlook 
which have led to the mathematical theory of motion. 
Let us, however, first try to state the objection more 

If the mathematical theory is adequate, nothing 
happens when a body moves except that it is in different 
places at different times. But in this sense the hour- 
hand and the second-hand are equally in motion, yet 
in the second-hand there is something perceptible to 
our senses which is absent in the hour-hand. We can 
see, at each moment, that the second-hand is moving, 
which is different from seeing it first in one place and 
then in another. This seems to involve our seeing 


it simultaneously in a number of places, although it 
must also involve our seeing that it is in some of these 
places earlier than in others. If, for example, I 
move my hand quickly from left to right, you seem 
to see the whole movement at once, in spite of the 
fact that you know it begins at the left and ends at 
the right. It is this kind of consideration, I think, 
which leads Bergson and many others to regard a 
movement as really one indivisible whole, not the 
series of separate states imagined by the mathe- 

To this objection there are three supplementary 
answers, physiological, psychological, and logical. We 
will consider them successively. 

(i) The physiological answer merely shows that, if 
the physical world is what the mathmatician supposes, 
its sensible appearance may nevertheless be expected 
to be what it is. The aim of this answer is thus the 
modest one of showing that the mathematical account 
is not impossible as applied to the physical world ; it 
does not even attempt to show that this account is 
necessary, or that an analogous account applies in 

When any nerve is stimulated, so as to cause a 
sensation, the sensation does not cease instantaneously 
with the cessation of the stimulus, but dies away in a 
short finite time. A flash of lightning, brief as it is 
to our sight, is briefer still as a physical phenomenon : 
we continue to see it for a few moments after the light- 
waves have ceased to strike the eye. Thus in the 
case of a physical motion, if it is sufficiently swift, we 
shall actually at one instant see the moving body 
throughout a finite portion of its course, and not 
only at the exact spot where it is at that instant. 
Sensations, however, as they die away, grow gradually 



fainter ; tnus the sensation due to a stimulus which 
is recently past is not exactly like the sensation due 
to a present stimulus. It follows from this that, 
when we see a rapid motion, we shall not only see a 
number of positions of the moving body simultaneously, 
but we shall see them with different degrees of intensity 
the present position most vividly, and the others 
with diminishing vividness, mvril sensation fades 
away into immediate memory. This state of things 
accounts fully for the perception of motion. A motion 
is perceived, not merely inferred, when it is sufficiently 
swift for many positions to be sensible at one time ; 
and the earlier and later parts of one perceived motion 
are distinguished by the less and greater vividness of 
the sensations. 

This answer shows that physiology can account for 
our perception of motion. But physiology, in speaking 
of stimulus and sense-organs and a physical motion 
distinct from the immediate object of sense, is assuming 
the truth of physics, and is thus only capable of show- 
ing the physical account to be possible, not of showing 
it to be necessary. This consideration brings us to 
the psychological answer. 

(2) The psychological answer to our difficulty about 
motion is part of a vast theory, not yet worked out, 
and only capable, at present, of being vaguely outlined. 
We considered this theory in the third and fourth 
lectures ; for the present, a mere sketch of its applica- 
tion to our present problem must suffice. The world of 
physics, which was assumed in the physiological 
answer, is obviously inferred from what is given in 
sensation; yet as soon as we seriously consider what is 
actually given in sensation, we find it apparently very 
different from the world of physics. The question is 
thus forced upon us : Is the inference from sense to 


physics a valid one ? I believe the answer to be 
affirmative, for reasons which I suggested in the third 
and fourth lectures ; but the answer cannot be either 
short or easy. It consists, broadly speaking, in show- 
ing that, although the particles, points, and instants 
with which physics operates are not themselves 
given in experience, and are very likely not actually 
existing things, yet, out of the materials provided in 
sensation, together with other particulars structurally 
similar to these materials, it is possible to make logical 
constructions having the mathematical properties 
which physics assigns to particles, points, and instants. 
If this can be done, then all the propositions of physics 
can be translated, by a sort of dictionary, into proposi- 
tions about the kinds of objects which are given in 

Applying these general considerations to the case 
of motion, we find that, even within the sphere of 
immediate sense-data, it is necessary, or at any rate 
more consonant with the facts than any other equally 
simple view, to distinguish instantaneous states of 
objects, and to regard such states as forming a compact 
series. Let us consider a body which is moving swiftly 
enough for its motion to be perceptible, and long enough 
for its motion to be not wholly comprised in one 
sensation. Then, in spite of the fact that we see 
a finite extent of the motion at one instant, the extent 
which we see at one instant is different from that 
which we see at another. Thus we are brought 
back, after all, to a series of momentary views of the 
moving body, and this series will be compact, like 
the former physical series of points. In fact, though 
the terms of the series seem different, the mathematical 
character of the series is unchanged, and the whole 
mathematical theory of motion will apply to it verbatim. 


When we are considering the actual data of sensa- 
tion in this connection, it is important to realize that 
two sense-data may be, and must sometimes be, really 
different when we cannot perceive any difference 
between them. An old but conclusive reason for 
believing this was emphasized by PoincarS. 1 In all 
cases of sense-data capable of gradual change, we 
may find one sense-datum indistinguishable from 
another, and that other indistinguishable from a 
third, while yet the first and third are quite easily 
distinguishable. Suppose, for example, a person with 
his eyes shut is holding a weight in his hand, and 
someone noiselessly adds a small extra weight. If 
the extra weight is small enough, no difference will be 
perceived in the sensation. After a time, another 
small extra weight may be added, and still no change 
will be perceived; but if both extra weights had 
been added at once, it may be that the change would 
be quite easily perceptible. Or, again, take shades 
of colour. It would be easy to find three stuffs of 
such closely similar shades that no difference could be 
perceived between the first and second, nor yet between 
the second and third, while yet the first and third 
would be distinguishable. In such a case, the second 
shade cannot be the same as the first, or it would be 
distinguishable from the third ; nor the same as the 
third, or it would be distinguishable from the first. 
It must, therefore, though indistinguishable from 
both, be really intermediate between them. 

Such considerations as the above show that, although 
we cannot distinguish sense-data unless they differ 
by more than a certain amount, it is perfectly reason- 
able to suppose that sense-data of a given kind, such 

1 " Le continu math&natique/' Revue de Mttaphysique et fo 
Morale, vol. i. p. 29. 


as weights or colours, really form a compact series. 
The objections which may be brought from a psycho- 
logical point of view against the mathematical theory 
of motion are not, therefore, objections to this theory 
properly understood, but only to a quite unnecessary 
assumption of simplicity in the momentary object 
of sense. Of the immediate object of sense, in the case 
of a visible motion, we may say that at each instant 
it is in all the positions which remain sensible at that 
instant ; but this set of positions changes continuously 
from moment to moment, and is amenable to exactly 
the same mathematical treatment as if it were a mere 
point. When we assert that some mathematical 
account of phenomena is correct, all that we primarily 
assert is that something definable in terms of the crude 
phenomena satisfies our formulae ; and in this sense 
the mathematical theory of motion is applicable to 
the data of sensation as well as to the supposed par- 
ticles of abstract physics. 

There are a number of distinct questions which are 
apt to be confused when the mathematical con- 
tinuum is said to be inadequate to the facts of sense. 
We may state these, in order of diminishing generality, 
as follows : 

(a) Are series possessing mathematical con- 
tinuity logically possible ? 

(6) Assuming that they are possible logically, are 
they not impossible as applied to actual sense- 
data, because, among actual sense-data, there are 
no such fixed mutually external terms as are to 
be found, e.g. in the series of fractions ? 

(c) Does not the assumption of points and 
instants make the whole mathematical account 
fictitious ? 


() Finally, assuming that all these objections 
have been answered, is there, in actual empirical 
fact, any sufficient reason to believe the world of 
sense continuous ? 

Let us consider these questions in succession. 

(a) The question of the logical possibility of the 
mathematical continuum turns partly on the ele- 
mentary misunderstandings we considered at the 
beginning of the present lecture, partly on the possi- 
bility of the mathematical infinite, which will occupy 
our next two lectures, and partly on the logical form 
of the answer to the Bergsonian objection which we 
stated a few minutes ago. I shall say no more on 
this topic at present, since it is desirable first to com- 
plete the psychological answer. 

(6) The question whether sense data are composed of 
mutually external units is not one which can be decided 
by empirical evidence. It is often urged that, as a 
matter of immediate experience, the sensible flux is 
devoid of divisions, and is falsified by the dissections 
of the intellect. Now I have no wish to argue that 
this view is contrary to immediate experience : I wish 
only to maintain that it is essentially incapable of 
being proved by immediate experience. As we saw, 
there must be among sense-data differences so slight 
as to be imperceptible : the fact that sense-data are 
immediately given does not mean that their differences 
also must be immediately given (though they may 
be). Suppose, for example, a coloured surface on 
which the colour changes gradually so gradually 
that the difference of colour in two very neighbouring 
portions is imperceptible, while the difference between 
more widely separated portions is quite noticeable. 
The effect produced, in such a case, will be precisely 


that of " interpenetration," of transition which is 
not a matter of discrete units. And since it tends to 
be supposed that the colours, being immediate data, 
must appear different if they are different, it seems 
easily to follow that "interpenetration" must be 
the ultimately right account. But this does not follow. 
It is unconsciously assumed, as a premiss for a reductio 
ai dbsurdum of the analytic view, that, if A and B 
are immediate data, and A differs from B, then the 
fact that they differ must also be an immediate datum. 
It is difficult to say how this assumption arose, but I 
think it is to be connected with the confusion between 
" acquaintance " and " knowledge about." Acquaint- 
ance, which is what we derive from sense, does not, 
theoretically at least, imply even the smallest " know- 
ledge about," i.e. it does not imply knowledge of any 
proposition concerning the object with which we are 
acquainted. It is a mistake to speak as if acquaint- 
ance had degrees : there is merely acquaintance and 
non-acquaintance. When we speak of becoming 
" better acquainted," as for instance with a person, 
what we must mean is, becoming acquainted with 
more parts of a certain whole ; but the acquaintance 
with each part is either complete or non-existent. 
Thus it is a mistake to say that if we were perfectly 
acquainted with an object we should know all about 
it. " Knowledge about " is knowledge of proposi- 
tions, which is not involved necessarily in acquaint- 
ance with the constituents of the propositions. To 
know that two shades of colour are different is know- 
ledge about them ; hence acquaintance with the two 
shades does not in any way necessitate the knowledge 
that they are different. 

From what has just been said it follows that the 
nature of sense-data cannot be validly used to prove 


that they are not composed of mutually external 
units. It may be admitted, on the other hand, that 
nothing in their empirical character specially necessi- 
tates the view that they are composed of mutually 
external units. This view, if it is held, must be held 
on logical, not on empirical grounds. I believe that 
the logical grounds axe adequate to the conclusion. 
They rest, at bottom, upon the impossibility of ex- 
plaining complexity without assuming constituents. 
It is undeniable that the visual field, for example, 
is complex ; and so far as I can see, there is always 
self-contradiction in the theories which, while admitting 
this complexity, attempt to deny that it results from 
a combination of mutually external units. But to 
pursue this topic would lead us too far from our theme, 
and I shall therefore say no more about it at present. 

(c) It is sometimes urged that the mathematical 
account of motion is rendered fictitious by its assump^ 
tion of points and instants. Now there are here two 
different questions to be distinguished. There is the 
question of absolute or relative space and time, and 
there is the question whether what occupies space 
and time must be composed of elements which have 
no extension or duration. And each of these ques- 
tions in turn may take two forms, namely : (a) is 
the hypothesis consistent with the facts and with 
logic ? 08) is it necessitated by the facts or by logic ? 
I wish to answer, in each case, yes to the first form 
of the question, and no to the second. But in any 
case the mathematical account of motion will not be 
fictitious, provided a right interpretation is given 
to the words " point " and " instant." A few words 
on each alternative will serve to make this clear. 

Formally, mathematics adopts an absolute theory 
of space and time, ie. it assumes that, besides the 


things which are in space and time, there are also 
entities, called "points" and "instants/' which 
axe occupied by things. This view, however, though 
advocated by Newton, has long been regarded by 
mathematicians as merely a convenient fiction. There 
is, so far as I can see, no conceivable evidence either 
for or against it. It is logically possible, and it is 
consistent with the facts. But the facts are also 
consistent with the denial of spatial and temporal 
entities over and above things with spatial and tem- 
poral relations. Hence, in accordance with Occam's 
razor, we shall do well to abstain from either assuming 
or denying points and instants. This means, so far 
as practical working out is concerned, that we adopt the 
relational theory ; for in practice the refusal to assume 
points and instants has the same effect as the denial 
of them. But in strict theory the two are quite 
different, since the denial introduces an element of 
unverifiable dogma which is wholly absent when we 
merely refrain from the assertion. Thus, although 
we shall derive points and instants from things, we 
shall leave the bare possibility open that they may 
also have an independent existence as simple entities. 

We come now to the question whether the things 
in space and time are to be conceived as composed of 
elements without extension or duration, i.e. of elements 
which only occupy a point and an instant. Physics, 
formally, assumes in its differential equations that 
things consist of elements which occupy only a point 
at each instant, but persist throughout time. For 
reasons explained in Lecture IV, the persistence of 
things through time is to be regarded as the 
formal result of a logical construction, not as necessarily 
implying any actual persistence. The same motives, 
in fact, which lead to the division of things into point- 


particles, ought presumably to lead to their division 
into instant-particles, so that the ultimate formal 
constituent of the matter in physics will be a point- 
instant-particle. But such objects, as well as the 
particles of physics, are not data. The same economy 
of hypothesis, which dictates the practical adoption 
of a relative rather than an absolute space and time, 
also dictates the practical adoption of material elements 
which have a finite extension and duration. Since, 
as we saw in Lecture IV, points and instants can be 
constructed as logical functions of such elements, the 
mathematical account of motion, in which a particle 
passes continuously through a continuous series of 
points, can be interpreted in a form which assumes 
only elements which agree with our actual data in 
having a finite extension and duration. Thus, so 
far as the use of points and instants is concerned, the 
mathematical account of motion can be freed from 
the charge of employing fictions. 

(f} But we must now face the question : Is there, 
in actual empirical fact, any sufficient reason to 
believe the world of sense continuous ? The answer 
here must, I think, be in the negative. We may 
say that the hypothesis of continuity is perfectly 
consistent with the facts and with logic, and that it 
is technically simpler than any other tenable hypo- 
thesis. But since our powers of discrimination among 
very similar sensible objects are not infinitely precise, 
it is quite impossible to decide between different 
theories which only differ in regard to what is below 
the margin of dfecrimination. If, for example, a 
coloured surface which we see consists of a finite 
number of very small surfaces, and if a motion which 
we see consists, like a cinematograph, of a large finite 
number of successive positions, there will be nothing 


empirically discoverable to show that objects of sense 
are not continuous. In what is called experienced con- 
tinuity, such as is said to be given in sense, there is a 
large negative element: absence of perception of 
difference occurs in cases which are thought to give 
perception of absence of difference. When, for 
example, we cannot distinguish a colour A from a 
colour B, nor a colour B from a colour C, but can 
distinguish A from C, the indistinguishabiHty is a 
purely negative fact, namely, that we do not perceive 
a difference. Even in regard to immediate data, 
this is no reason for denying that there is a difference. 
Thus, if we see a coloured surface whose colour changes 
gradually, its sensible appearance if the change is 
continuous will be indistinguishable from what it 
would be if the change were by small finite jumps. If 
this is true, as it seems to be, it follows that there can 
never be any empirical evidence to demonstrate that 
the sensible world is continuous, and not a collection 
of a very large finite number of dements of which each 
differs from its neighbour in a finite though very small 
degree. The continuity of space and time, the infinite 
number of different shades in the spectrum, and so 
on, are all in the nature of unverifiable hypotheses 
perfectly possible logically, perfectly consistent 
with the known facts, and simpler technically than 
any other tenable hypotheses, but not the sole hypo- 
theses which are logically and empirically adequate. 

If a relational theory of instants is constructed, in 
which an " instant " is defined as a group of events 
simultaneous with each other and not all simultaneous 
with any event outside the group, then if our resulting 
series of instants is to be compact, it must be possible, 
if x wholly precedes y , to find an event z, simultaneous 
with part of x, which wholly precedes some event 


which wholly precedes y. Now this requires that the 
number of events concerned should be infinite in any 
finite period of time. If this is to be the case in the 
world of one man's sense-data, and if each sense- 
datum is to have not less than a certain finite temporal 
extension, it will be necessary to assume that we always 
have an infinite number of sense-data simultaneous 
with any given sense-datum. Applying similar con- 
siderations to space, and assuming that sense-data 
are to have not less than a certain spatial extension, 
it will be necessary to suppose that an infinite number 
of sense-data overlap spatially with any given sense- 
datum. This hypothesis is possible, if we suppose a 
single sense-datum, e.g. in sight, to be a finite surface, 
enclosing other surfaces which are also single sense- 
data. But there are difficulties in such a hypothesis, 
and I do not think that these difficulties could be 
successfully met. If they cannot, we must do one 
of two things : either declare that the world of one 
man's sense-data is not continuous, or else refuse to 
admit that there is any lower limit to the duration 
and extension of a single sense-datum. The latter 
hypothesis seems untenable, so that we are apparently 
forced to conclude that the space of sense-data is 
not continuous ; but that does not prevent us from 
admitting that sense-data have parts which are not 
sense-data, and that the space of these parts may be 
continuous. The logical analysis we have been con- 
sidering provides the apparatus for dealing with the 
various hypotheses, and the empirical decision between 
them is a problem for the psychologist. 

(3) We have now to consider the logical answer to the 
alleged difficulties of the mathematical theory of 
motion, or rather to the positive theory which is 
urged on the other side. The view urged explicitly 


by Bergson, and implied in the doctrines of many 
philosophers, is, that a motion is something indivisible, 
not validly analysable into a series of states. This is 
part of a much more general doctrine, which holds 
that analysis always falsifies, because the parts of a 
complex whole are different, as combined in that whole, 
from what they would otherwise be. It is very difficult 
to state this doctrine in any form which has a precise 
meaning. Often arguments are used which have no 
bearing whatever upon the question. It is urged, 
for example, that when a man becomes a father, 
his nature is altered by the new relation in which he 
finds himself, so that he is not strictly identical with 
the man who was previously not a father. This may 
be true, but it is a causal psychological fact, not a 
logical fact. The doctrine would require that a man 
who is a father cannot be strictly identical with a 
man who is a son, because he is modified in one way 
by the relation of fatherhood and in another by that 
of sonship. In fact, we may give a precise statement 
of the doctrine we are combating in the form : There 
can never be two facts concerning the same thing. A 
fact concerning a thing always is or involves a relation 
to one or more entities ; thus two facts concerning the 
same thing would involve two relations of the same 
thing. But the doctrine in question holds that a thing 
is so modified by its relations that it cannot be the same 
in one relation as in another. Hence, if this doctrine 
is true, there can never be more than one fact con- 
cerning any one thing. I do not think the philosophers 
in question have realized that this is the precise state- 
ment of the view they advocate, because in this form 
the view is so contrary to plain truth that its falsehood 
is evident as soon as it is stated. The discussion of 
this question, however, involves so many logical 


subtleties, and is so beset with difficulties, that I shall 
not pursue it further at present. 

When once the above general doctrine is rejected, 
it is obvious that, where there is change, there must 
be a succession of states. There cannot be change 
and motion is only a particular case of change unless 
there is something different at one time from what 
there is at some other time. Change, therefore, must 
involve relations and complexity, and must demand 
analysis. So long as our analysis has only gone as 
far as other smaller changes, it is not complete ; if 
it is to be complete, it must end with terms that are 
not changes, but are related by a relation of earlier 
and later. In the case of changes which appear 
continuous, 'such as motions, it seems to be impos- 
sible to find anything other than change so long as 
we deal with finite periods of time, however short. 
We are thus driven back, by the logical necessities 
of the case, to the conception of instants without 
duration, or at any rate without any duration which 
even the most delicate instruments can reveal. This 
conception, though it can be made to seem difficult, is 
really easier than any other that the facts allow. It 
is a kind of logical framework into which any tenable 
theory must fit not necessarily itself the statement 
of the crude facts, but a form in which statements 
which are true of the crude facts can be i&ade by a 
suitable interpretation. The direct consideration of 
the crude facts of the physical world has been under- 
taken in earlier lectures ; in the present lecture, we 
have only been concerned to show that nothing in 
the crude facts is inconsistent with the mathematical 
doctrine of continuity, or demands a continuity of 
a radically different kind from that of mathematical 



IT will be remembered that, when we enumerated the 
grounds upon which the reality of the sensible world 
has been questioned, one of those mentioned was the 
supposed impossibility of infinity and continuity. In 
view of our earlier discussion of physics, it would seem 
that no conclusive empirical evidence exists in favour 
of infinity or continuity in objects of sense or in matter. 
Nevertheless, the explanation which assumes infinity 
and continuity remains incomparably easier and more 
natural, from a scientific point of view, than any other, 
and since Georg Cantor has shown that the supposed 
contraditions are illusory, there is no longer any reason 
to struggle after a finitist explanation of the world. 

The supposed difficulties of continuity all have their 
source in the fact that a continuous series must have 
an infinite number of terms, and are in fact difficulties 
concerning infinity. Hence, in freeing the infinite 
from contradiction, we are at the same time showing 
the logical possibility of continuity as assumed in 

The kind of way in which infinity has been used to 
discredit the world of sense may be illustrated by 
Kant's first two antinomies. In the first, the thesis 
states : " The world has a beginning in time, and as 


regards space is enclosed within limits " ; the anti- 
thesis states : " The world has no beginning and no 
limits in space, but is infinite in respect of both time 
and space." Kant professes to prove both these 
propositions, whereas, if what we have said on modern 
logic has any truth, it must be impossible to prove 
either. In order, however, to rescue the world of 
sense, it is enough to destroy the proof of one of 
the two. For our present purpose, it is the proof that 
the world is finite that interests us. Kant's argument 
as regards space here rests upon his argument as 
regards time. We need therefore only examine the 
argument as regards time. What he says is as follows : 

" For let us assume that the world has no beginning 
as regards time, so that up to every given instant an 
eternity has elapsed, and therefore an infinite series of 
successive states of the things in the world has passed 
by. But the infinity of a series consists just in this, 
that it can never be completed by successive syn- 
thesis. Therefore an infinite past world-series is 
impossible, and accordingly a .beginning of the world 
is a necessary condition of its existence ; which was 
the first thing to be proved." 

Many different criticisms might be passed on this 
argument, but we will content ourselves with a bare 
TninjTnnTn , To begin with, it is a mistake to define 
the infinity of a series as " impossibility of completion 
by successive synthesis." The notion of infinity, as 
we shall see in the next lecture, is primarily a property 
of dosses, and only derivatively applicable to series ; 
classes which are infinite are given all at once by the 
defining properly of their members, so that there is 
no question of " completion " or of " successive syn- 
thesis." And the word "synthesis," by suggesting 
the mental activity of synthesizing, introduces, more 


or less surreptitiously, that reference to mind by which 
all Kant's philosophy was infected. In the second 
place, when Kant says that an infinite series can 
" never " be completed by successive synthesis, all 
that he has even conceivably a right to say is that it 
cannot be completed in a finite time. Thus what he 
really proves is, at most, that if the world had no 
beginning, it must have already existed for an infinite 
time. This, however, is a very poor conclusion, by 
no means suitable for his purposes. And with this 
result we might, if we chose, take leave of the first 

It is worth while, however, to consider how Kant 
came to make such an elementary blunder. What 
happened in his imagination was obviously something 
like this : Starting from the present and going back- 
wards in time, we have, if the world had no beginning, 
an infinite series of events. As we see from the word 
"synthesis," he imagined a mind trying to grasp 
these successively, in the reverse order to that in 
which they had occurred, i.e. going from the present 
backwards. This series is obviously one which has 
no end. But the series of events up to the present 
has an end, since it ends with the present. Owing to 
the inveterate subjectivism of his mental habits, he 
failed to notice that he had reversed the sense of the 
series by substituting backward synthesis for forward 
happening, and thus he supposed that it was necessary 
to identify the mental series, which had no end, with 
the physical series, which had an end but no beginning. 
It was this mistake, I think, which, operating 
unconsciously, led M to attribute validity to a singu- 
larly flimsy piece of fallacious reasoning. 

The second antimony illustrates the dependence of 
the problem of continuity upon that of infinity. The 



thesis states : " Every complex substance in the 
world consists of simple parts, and -there exists every- 
where nothing but the simple or what is composed 
of it." The antithesis states : " No complex thing 
in the world consists of simple parts, and everywhere in 
it there exists nothing simple." Here, as before, the 
proofs of both thesis and antithesis are open to criti- 
cism, but for the purpose of vindicating physics and 
the world of sense it is enough to find a fallacy in 
one of the proofs. We will choose for this purpose 
the proof of the antithesis, which begins as follows : 

" Assume that a complex thing (as substance) con- 
sists of simple parts. Since all external relation, and 
therefore all composition out of substances, is only 
possible in space, the space occupied by a complex 
thing must consist of as many parts as the thing con- 
sists of. Now space does not consist of simple parts, 
but of spaces." 

The rest of his argument need not concern us, for 
the nerve of the proof lies in the one statement : 
" Space does not consist of simple parts, but of spaces." 
This is like Bergson's objection to " the absurd pro- 
position that motion is made up of immobilities." 
Kant does not tell us why he holds that a space must 
consist of spaces rather than of simple parts. Geo- 
metry regards space as made up of points, which are 
simple ; and although, as we have seen, this view is 
not scientifically or logically necessary, it remains 
prima facie possible, and its mere possibility is enough 
to vitiate Kant's argument. For, if his proof of the 
thesis of the antinomy were valid, and if the antithesis 
could only be avoided by assuming points, then the 
antinomy itself would afford a conclusive reason in 
favour of points. Why, then, did Kant tTiinlr it im- 
possible that space should be composed of points ? 


I think two considerations probably influenced him. 
In the first place, the essential thing about space is 
spatial order, and mere points, by themselves, will 
not account for spatial order. It is obvious that his 
argument assumes absolute space ; but it is spatial 
relations that are alone important, and they cannot 
be reduced to points. This ground for his view 
depends, therefore, upon his ignorance of the logical 
theory of order and his oscillations between absolute 
and relative space. But there is also another ground 
for his opinion, which is more relevant to our present 
topic. This is the ground derived from infinite divisi- 
bility. A space may be halved, and then halved again, 
and so on ad infinitwn, and at every stage of the pro- 
cess the parts are still spaces, not points. In order 
to reach points by such a method, it would be necessary 
to come to the end of an unending process, which is 
impossible. But just as an infinite class can be given 
all at once by its defining concept, though it cannot be 
reached by successive enumeration, so an infinite 
set of points can be given all at once as making up a 
line or area or volume, though they can never be 
reached by the process of successive division. Thus the 
infinite divisibility of space gives no ground for deny- 
ing that space is composed of points. Kant does not 
give his grounds for this denial, and we can therefore 
only conjecture what they were. But the above 
two grounds, which we have seen to be fallacious, 
seem sufficient to account for his opinion, and we may 
therefore conclude that the antithesis of the second 
antinomy is unproved. 

The above illustration of Kant's antinomies has 
only been introduced in order to show the relevance 
of the problem of infinity to the problem of the reality 
of objects of sense. In the remainder of the present 


lecture, I wish to state and explain the problem of 
infinity, to show how it arose, and to show the irrele- 
vance of all the solutions proposed by philosophers. 
In the following lecture, I shall try to explain the 
true solution, which has been discovered by the 
mathematicians, but nevertheless belongs essentially 
to philosophy. The solution is definitive, in the sense 
that it entirely satisfies and convinces all who study it 
carefully. For over two thousand years the human 
intellect was baffled by the problem ; its many failures 
and its ultimate success maie this problem peculiarly 
apt for the illustration of method. 

The problem appears to have first arisen in some such 
way as the following. 1 Pythagoras and his followers, 
who were interested, like Descartes, in the application 
gf number to geometry, adopted in that science more 
arithmetical methods than those with which Euclid 
has made us familiar. They, or their contemporaries 
the atomists, believed, apparently, that space is com- 
posed of indivisible points, while time is composed 
of indivisible instants.* This belief would not, by 
itself, have raised the difficulties which they encoun- 
tered, but it was presumably accompanied by another 
belief, that the number of points in any finite area 
or of instants in any finite period must be finite. I 
do not suppose that this latter belief was a conscious 
one, because probably no other possibility had occurred 
to them. But the belief nevertheless operated, and 

* In what concerns the early Greek philosophers, my 
knowledge is largely derived from Burnet's valuable work, 
Early Greek Philosophy (2nd ed., London, 1908). I have also 
been greatly assisted by Mr. D. S. Robertson of Trinity College, 
who has supplied the deficiencies of my knowledge of Greek, 
and brought important references to my notice. 

* Cf . Aristotle, Metaphysics, M. 6, 10806, 18 sqq., and 
10836, 8 sqq. 


very soon brought them into conflict with facts which 
they themselves discovered. Before explaining how 
this occurred, however, it is necessary to say one word 
in explanation of the phrase " finite number." The 
exact explanation is a matter for our next lecture ; for 
the present, it must suffice to say that I mean o and 
I and 2 and 3 and so one, for ever in other words, 
any number that can be obtained by successively 
adding ones. This includes all the numbers that 
can be expressed by means of our ordinary numerals, 
and since such numbers can be made greater and 
greater, without ever reaching an unsurpassable 
maximum, it is easy to suppose that there are no other 
numbers. But this supposition, natural as it is, is 

Whether the Pythagoreans themselves believed 
space and time to be composed of indivisible points 
and instants is a debatable question. 1 It would seem 
that the distinction between space -and matter had 

z There is some reason to think that the Pythagoreans . 
distinguished between discrete and continuous quantity. 
G. J. Allman, in his Greek Geometry from Tholes to Euclid. 
says (p. 23) : " The Pythagoreans made a fourfold division of 
mathematical science, attributing one of its parts to the how 
many, ?6 ir6aov, and the other to the how much, r6 tn\\lKw ; 
and they assigned to each of these parts a twofold division. 
For they said that discrete quantity, or the how many, either 
subsists by itself or must be considered with relation to some 
other ; but that continued quantity, or the how much, is either 
stable or in motion. Hence they affirmed that arithmetic 
contemplates that discrete quantity which subsists by itself, 
but music that which is related to another ; and that geometry 
considers continued quantity so far as it is immovable ; but 
astronomy (rip afaupucfa)- contemplates continued quantity 
so far as it is of a self-motive nature. (Proclus, ed. Friedlein, 
p. 35. As to the distinction between rd mjMxov, continuous, 
and TO ir6oov, discrete quantity, see Iambi., in Nicomachi 
Geyaseni Arithmeticam introductionem, ed. Tennulius, p. 148.) " 

a. P . 4 8- 


not yet been clearly made, and that therefore, when 
an atomistic view is expressed, it is difficult to decide 
whether particles of matter or points of space are 
intended. There is an interesting passage z in 
Aristotle's Physics* where he says : 

" The Pythagoreans all maintained the existence of 
the void, and said that it enters into the heaven itself 
from the boundless breath, inasmuch as the heaven 
breathes in the void also ; and the void differentiates 
natures, as if it were a sort of separation of consecu- 
tives, and as if it were their differentiation ; and that 
this also is what is first in numbers, for it is the void 
which differentiates them." 

This seems to imply that they regarded matter as 
consisting of atoms with empty space in between. 
But if so, they must have thought space could be 
studied by only paying attention to the atoms, for 
otherwise it would be hard to account for their arith- 
metical methods in geometry, or for their statement 
that " things are numbers." 

The difficulty which beset the Pythagoreans in 
their attempts to apply numbers arose through their 
discovery of incommensurables, and this, in turn, arose 
as follows. Pythagoras, as we all learnt in youth, 
discovered the proposition that the sum of the squares 
on the sides of a right-angled triangle is equal to the 
square on the hypotenuse. It is said that he sacrificed 
an ox when he discovered this theorem ; if so, the ox 
was the first martyr to science. But the theorem, 
though it has remained his chief claim to immortality, 
was soon found to have a consequence fatal to his 

1 Referred to by Burnet, op. dt., p. 120. 

iv. f 6. 213$, 22 ; H. Ritter and L. Preller, Historia Philo- 
sophies Gr&ca, 8th ed. f Gotha, 1898, p. 75 (this work will be 
referred to in future as " R. P."). 


whole philosophy. Consider the case of a right-angled 
triangle whose two sides are equal, such a triangle 
as is formed by two sides of a square and a diagonal. 
Here, in virtue of the theorem, the square on the 
diagonal is double of the square on either of the sides. 
But Pythagoras or his early followers easily proved 
that the square of one whole number cannot be double 
of the square of another. 1 Thus the length of the 
side and the length of the diagonal are incommen- 
surable ; that is to say, however small a unit of length 
you take, if it is contained an exact number of times 
in the side, it is not contained any exact number of 
times in the diagonal, and vice versa. 

Now this fact might have been assimilated by sortie 
philosophies without any great difficulty, but to the 
philosophy of Pythagoras it was absolutely fatal. 
Pythagoras held that number is the constitutive 
essence of all things, yet no two numbers could express 
the ratio of the side of a square to the diagonal. It 
would seem probable that we may expand his difficulty, 
without departing from his thought, by assuming that 
he regarded the length of a line as determined by the 
number of atoms contained in it a line two inches 
long would contain twice as many atoms as a line 
one inch long, and so on. But if this were the truth, 
then there must be a definite numerical ratio between 

1 The Pythagorean proof is roughly as follows. If possible, 
let the ratio of the diagonal to the side of a square be m/n, 
where m and n are whole numbers having no common factor. 
Then we must have w a = 2n*. Now the square of an odd 
number is odd, but m a , being equal to 2, is even. Hence 
m must be even. But the square of an even number divides 
by 4, therefore *, which is half of m*, must be even. There- 
fore n must be even. But, since m is even, and m and n have 
no common factor, n must be odd. Thus n must be both odd 
and even, which is impossible; and therefore the diagonal 
and the side cannot have a rational ratio. 


any two finite lengths, because it was supposed that 
the number of atoms in each, however large, must be 
finite. Here there was an insoluble contradiction. 
The Pythagoreans, it is said, resolved to keep the 
existence of incommensurables a profound secret, 
revealed only to a few of the supreme heads of the 
sect ; and one of their number, Hippasos of Meta- 
pontion, is even said to have been shipwrecked at sea 
for impiously disclosing the terrible discovery to 
their enemies. It must be remembered that Pytha- 
goras was the founder of a new religion as well as the 
teacher of a new science : if the science came to be 
doubted, the disciples might fall into sin, and perhaps 
even eat beans, which according to Pythagoras is as 
bad as eating parents' bones. 

The problem first raised by the discovery of incom- 
mensurables proved, as time went on, to be one of the 
most severe and at the same time most far-reaching 
problems that have confronted the human intellect 
in its endeavour to understand the world. It showed 
at once that numerical measurement of lengths, if it 
was to be made accurate, must require an arithmetic 
more advanced and more difficult than any that the 
ancients possessed. They therefore set to work to 
reconstruct geometry on a basis which did not assume 
the universal possibility of numerical measurement 
a reconstruction which, as may be seen in Euclid, 
they effected with extraordinary skill and with great 
logical acumen. The moderns, under the influence 
of Cartesian geometry, have reasserted the universal 
possibility of numerical measurement, extending arith- 
metic, partly for that purpose, so as to include what 
are called " irrational " numbers, which give the ratios 
of incommensurable lengths. But although irrational 
numbers have long been used without a qualm, it is 


only in quite recent years that logically satisfactory 
definitions of them have been given. With these 
definitions, the first and most obvious form of the 
difficulty which confronted the Pythagoreans has 
been solved ; but other forms of the difficulty remain 
to be considered, and it is these that introduce us 
to the problem of infinity in its pure form. 

We saw that, accepting the view that a length is 
composed of points, the existence of incommensurables 
proves that every finite length must contain an infinite 
number of points. In other words, if we were to take 
away points one by one, we should never have taken 
away all the points, however long we continued the 
process. The number of points therefore, cannot be 
counted, for counting is a process which enumerates 
things one by one. The property of being unable to 
be counted is characteristic of infinite collections, and 
is a source of many of their paradoxical qualities. 
So paradoxical are these qualities that until our own 
day they were thought to constitute logical contra- 
dictions. A long line of philosophers, from Zeno x 
to M. Bergson, have based much of their metaphysics 
upon the supposed impossibility of infinite collections. 
Broadly speaMng, the difficulties were stated by Zeno, 
and nothing material was added until we reach Bol- 
zano's Paradoxien des Unendlichlen, a little work 
written .in 1847-8, and published posthumously in 
1851. Intervening attempts to deal with the problem 
are futile and negligible. The definitive solution oi 
the difficulties is due, not to Bolzano, but to Georg 
Cantor, whose work on this subject first appeared in 

1 In regard to Zeno and the Pythagoreans, I have derived 
much valuable information and criticism from Mr. P. . B 


In order to understand Zeno, and to realize how 
little modern orthodox metaphysics has added to the 
achievements of the Greeks, we must consider for a 
moment his master Pannenides, in whose interest 
the paradoxes were invented. 1 Parmenides expounded 
his views in a poem divided into two parts, called 
" the way of truth " and " the way of opinion " 
like Mr. Bradley's "Appearance" and "Reality," 
except that Paxmenides tells us first about reality 
and then about appearance. " The way of opinion," 
in his philosophy, is, broadly speaking, Pytha- 
goreanism ; it begins with a warning : " Here I shall 
close my trustworthy speech and thought about the 
truth. Henceforward learn the opinions of mortals, 
giving ear to the deceptive ordering of my words." 
What has gone before has been revealed by a goddess, 
who tells him what really is. Reality, she says, is 
uncreated, indestructible, unchanging, indivisible; it 
is " immovable in the bonds of mighty chains, without 
beginning and without end ; since coming into being 
and passing away have been driven afar, and true 
belief has cast them away." The fundamental prin- 
ciple of his inquiry is stated in a sentence which would 
not be out of place in Hegel : " Thou canst not know 
what is not that is impossible nor utter it ; for it 
is the same thing that can be thought and that can 
be." And again : " It needs must be that what can 
be thought and spoken of is ; for it is possible for it 
to be, and it is not possible for what is nothing to be." 
The impossibility of change follows from this principle ; 

So Plato makes Zeno say in the Parmenides, apropos of 
his philosophy as a whole; and all internal and external 
evidence supports this view. 

* " With Parmenides," Hegel says, " philosophizing proper 
began." Werke (edition of 1840), vol. xiii. p. 274. 


for what is past can be spoken of, and therefore, by 
the principle, still is. 

The great conception of a reality behind the passing 
illusions of sense, a reality one, indivisible, and un- 
changing, was thus introduced into Western philosophy 
by Parmenides, not, it would seem, for mystical or 
religious reasons, but on the basis of a logical argument 
as to the impossibility of not-being. All the great 
metaphysical systems notably those of Plato, Spinoza, 
and Hegel are the outcome of this fundamental 
idea. It is difficult to disentangle the truth and the 
error in this view. The contention that time is unreal 
and that the world of sense is illusory must, I think, 
be regarded as based upon fallacious reasoning. 
Nevertheless, there is some sense easier to fed than 
to state in which time is an unimportant and super- 
ficial characteristic of reality. Past and future must 
be acknowledged to be as real as the present, and a 
certain emancipation from slavery to time is essential 
to philosophic thought. The importance of time is 
rather practical than theoretical, rather in relation 
to our desires than in relation to truth. A truer 
image of the world, I think, is obtained by picturing 
things as entering into the stream of time from an 
eternal world outside, than from a view which regards 
time as the devouring tyrant of all that is. Both in 
thought and in feeling, to realize the unimportance of 
time is the gate of wisdom. But unimportance is 
not unreality ; and therefore what we shall have to 
say about Zeno's arguments in support of Parmenides 
must be mainly critical 

The relation of Zeno to Parmenides is explained 
by Plato x in the dialogue in which Socrates, as a young 
man, learns logical acumen and philosophic dis- 
Parmenides, 128 A-D. 


interestedness from their dialectic. I quote from 
Jowett's translation : 

"I see, Parmenides, said Socrates, that Zeno is 
your second self in his writings too; he puts what 
you say in another way, and would fain deceive us 
into believing that he is telling us what is new. For 
you, in your poems, say All is one, and of this you 
adduce excellent proofs ; and he on the other hand 
says There is no Many ; and on behalf of this he offers 
overwhelming evidence. To deceive the world, as 
you have done, by saying the same thing in different 
ways, one of you affirming the one, and the other 
denying the many, is a strain of art beyond the reach 
of most of us. 

" Yes, Socrates, said Zeno. But although you are 
as keen as a Spartan hound in pursuing the track, 
you do not quite apprehend the true motive of the 
composition, which is not really such an ambitious 
work as you imagine ; for what you speak of was an 
accident ; I had no serious intention of deceiving the 
world. The truth is that these writings of mine were 
meant to protect the arguments of Parmenides against 
those who scoff at him and show the many ridiculous 
and contradictory results which they suppose to 
follow from the affirmation of the one. My answer 
is an address to the partisans of the many, whose 
attack I return with interest by retorting upon them 
that their hypothesis of the being of the many if 
carried out appears in a still more ridiculous light than 
the hypothesis of the being of the one." 

Zeno's four arguments against motion were intended 
to exhibit the contradictions that result from supposing 
that there is such a thing as change, and thus to support 
the Parmenidean doctrine that reality is unchanging. 1 

' This interpretation is combated by Milhaud, Les philo- 


Unfortunately, we only know his arguments through 
Aristotle, 1 who stated them in order to refute them. 
Those philosophers in the present day who have had 
their doctrines stated by opponents will realize that 
a just or adequate presentation of Zeno's position is 
hardly to be expected from Aristotle ; but by some 
care in interpretation it seems possible to reconstruct 
the so-called " sophisms " which have been " refuted " 
by every tyro from that day to this. 

Zeno's arguments would seem to be " ad hominem " ; 
that is to say, they seem to assume premisses granted 
by his opponents, and to show that, granting these 
premisses, it is possible to deduce consequences which 
his opponents must deny. In order to decide whether 
they are valid arguments or " sophisms," it is necessary 
to guess at the tacit premisses, and to decide who was 
the " homo " at whom they were aimed. Some main- 
tain that they were aimed at the Pythagoreans, 3 
while others have held that they were intended to 
refute the atomists.s M. Evellin, on the contrary, 
holds that they constitute a refutation of infinite 
divisibility^ while M. G. Noel, in the interests of 
Hegel, maintains that the first two arguments refute 

sophes-gtometrcs da la Greet, p. 140 n. ( but his reasons do not 
seem to me convincing. All the interpretations in what 
follows are open to question, but all have the support of 
reputable authorities. 

Physics, vi. 9. 2396 (R-P- 136-139). 

* Cf . Gaston Milhaud, Les philosophes-g&m&tres de la Grece t 
p. 140 n. ; Paul Tannery, Pour I'histoire de la science hellene, 
p. 249 ; Buniet, op. tit., p. 362. 

s Cf. R. K. Gaye, "On Aristotle, Physics, Z be." Journal 
of Philology, vol. xxxi. esp. p. HI. Also Moritz Cantor, 
Vorlesungen fiber Geschichte der Mathematik, ist ed. f vol. i., 
1880, p. 168, who, however, subsequently adopted Paul 
Tannery's opinion, Vorlesungen, 3rd ed. (vol. i. p. 200). 

4 " Le mouvement et les partisans des indivisibles," Re we 
de MJtaphysique et de Morale, vol. i. pp. 382-395- 


infinite divisibility, while the next two refute indi- 
visibles. 1 Amid such a bewildering variety of inter- 
pretations, we can at least not complain of any 
restrictions on our liberty of choice. 

The historical questions raised by the above-men- 
tioned discussions are no doubt largely insoluble, owing 
to the very scanty material from which our evidence 
is derived. The points which seem fairly clear axe 
the following : (i) That, in spite of MM. Milhaud and 
Paul Tannery, Zeno is anxious to prove that motion 
is really impossible, and that he desires to prove this 
because he follows Parmenides in denying plurality ; a 
(2) that the third and fourth arguments proceed on 
the hypothesis of indivisibles, a hypothesis which, 
whether adopted by the Pythagoreans or not, was 
certainly much advocated, as may be seen from the 
treatise On Indivisible Lines attributed to Aristotle. 
As regards the first two arguments, they would seem 
to be valid on the hypothesis of indivisibles, and also, 
without this hypothesis, to be such as would be valid 
if the traditional contradictions in infinite numbers 
were insoluble, which they are not. 

We may conclude, therefore, that Zeno's polemic 
is directed against the view that space and time 
consist of points and instants ; and that as against the 
view that a finite stretch of space of time consists of 
a finite number of points and instants, his arguments 
are not sophisms, but perfectly valid. 

The conclusion which Zeno wishes us to draw is that 
plurality is a delusion, and spaces and times are really 
indivisible. The other conclusion which is possible, 

' "Le mouvcinent et les arguments de Z6non d'filde," 
Revue de Mttaphysique et de Morale, vol. i. pp. 107-125. 

Gf . N. Brochard, " Les prftendus sophismes de Z6non 
d'filee, " Revue de MJtaphysique et de Morale, vol. i. pp. 209-215. 


namely that the number of points and instants is 
infinite, was not tenable so long as the infinite was 
infected with contradictions. In a fragment which 
is not one of the four famous arguments against motion, 
Zeno says : 

" If things are a many, they must be just as many 
as they are, and neither more nor less. Now, if 
they are as many as they are, they wi]l be finite in 

" If things are a many, they will be infinite in 
number ; for there will always be other things between 
them, and others again between these. And so things 
are infinite in number." * 

This argument attempts to prove that, if there are 
many things, the number of them must be both finite 
and infinite, which is impossible ; hence we are to 
conclude that there is only one thing. But the weak 
point in the argument is the phrase : " If they are 
just as many as they axe, they will be finite in number." 
This phrase is not very dear, but it is plain that it 
assumes the impossibility of definite infinite numbers. 
Without this assumption, which is now known to 
be false, the arguments of Zeno, though they suffice 
(on certain very reasonable assumptions) to dispel 
the hypothesis of finite indivisibles, do not suffice to 
prove that motion and change and plurality are im- 
possible. They are not, however, on any view, mere 
foolish quibbles : they are serious arguments, raising 
difficulties which it has taken two thousand years to 
answer, and which even now are fatal to the teachings 
of most philosophers. 

The first of Zeno's arguments is the argument of 

i Simplicius, Phys., 140, 28 D (R.P. 133) ; Biirnet, op. cit. f 
pp. 364-365- 


the race-course, which is paraphrased by Burnet as 
follows : x 

" You cannot get to the end of a race-course. You 
cannot traverse an infinite number of points in a finite 
time. You must traverse the half of any given dis- 
tance before you traverse the whole, and the half of 
that again before you can traverse it. This goes on 
ad infinitum, so that there are an infinite number of 
points in any given space, and you cannot touch an 
infinite number one by one in a finite time." a 

Zeno appeals here, in the first place, to the fact that 
any distance, however small, can be halved. From 
this it follows, of course, that there must be an infinite 
number of points in a line. But Aristotle represents 

' Op. cit., p. 367. 

Aristotle's words are : " The first is the one on the non- 
existence of motion on the ground that what is moved must 
always attain, the middle point sooner than the end-point, on 
which we gave our opinion in the earlier part of our discourse." 
Phys., vi 9. 9398 (R.P. 136). Aristotle seems to refer to 
Phys., vi. 2. 223AB [R-P. I3*>A] .: " All space is continuous, 
for t and space are divided into the same and equal divisions. 
. . . Wherefore also Zeno's argument is fallacious, that it is 
sible to go through an infinite collection or to touch an 

infinite collection one by one in a finite time. For there are 
two senses in which the term ' infinite ' is applied both to 
length and to time, and in fact to all continuous things, either 
in regard to divisibility, or in regard to the ends. Now it is 
not possible to touch things infinite in regard to number in a 
finite time, but it is possible to touch things infinite in regard 
to divisibility: for time itself also is infinite in this sense. 
So that in fact we go through an infinite [space], in an infinite 
[time] and not in a finite [time], and we touch infinite things 
with infinite things, not with finite things." Philoponus, a 
sixth-century commentator (R.P. I36A, Ex&. Paris Philop. in 
Arist. Phys., 803, 2. Vit.), gives the following illustration: 
" For if a thing were moved the space of a cubit in one hour, 
since in every space there are an infinite number of points, 
the thing moved must needs touch all the points of the space : 
it will then go through an infinite collection in a finite time, 
which is impossible." 


him as arguing, you cannot touch an infinite number of 
points one by one in a finite time. The words " one 
by one " are important, (i) If att the points touched 
are concerned, then, though you pass through them 
continuously, you do not touch them " one by one/' 
That is to say, after touching one, there is not another 
which you touch next : no two points are next each 
other, but between any two there are always an infinite 
number of others, which cannot be enumerated one 
by one. (2) If, on the other hand, only the successive 
middle points are concerned, obtained by always 
halving what remains of the course, then the points 
are reached one by one, and, though they are infinite 
in number, they are in fact all reached in a finite 
time. His argument to the contrary may be supposed 
to appeal to the view that a finite time must consist 
of a finite number of instants, in which case what he 
says would be perfectly true on the assumption that 
the possibility of continued dichotomy is undeniable. 
If, on the other hand, we suppose the argument 
directed against the partisans of infinite divisibility, 
we must suppose it to proceed as follows : x " The 
points given by successive halving of the distances 
still to be traversed are infinite in number, and are 
reached in succession, each being reached a finite 
time later than its predecessor ; but the sum of an 
infinite number of finite times must be infinite, and 
therefore the process will never be completed/* It is 
very possible that this is historically the right inter- 
pretation, but in this form the argument is invalid. 
If half the course takes half a minute, and the next 
quarter takes a quarter of a minute, and so on, the 
whole course will take a minute. The apparent 

' a. Mr. C. D. Broad, " Note on Achilles and the Tortoise/ 
Mind. N.S,, vol. xxii. pp. 3iS-g. 



force of the argument, on this interpretation, lies 
solely in the mistaken supposition that there cannot 
be anything between the whole of an infinite series, 
which can be seen to be false by observing that i is 
beyond the whole of the infinite series, J, f, f , -J-$-, . . . 

The second of Zeno's arguments is the one concern- 
ing Achilles and the tortoise, which has achieved more 
notoriety than the others. It is paraphrased by 
Burnet as follows : I 

"Achilles will never overtake the tortoise. He 
must first reach the place from which the tortoise 
started. By that time the tortoise will have got 
some way ahead. Achilles must then make up 
that, and again the tortoise will be ahead. He 
is always coming nearer, but he never makes up 
to it." * 

This argument is essentially the same as the previous 
one. It shows that, if Achilles ever overtakes the 
tortoise, it must be after an infinite number of instants 
have elapsed since he started. This is in fact true ; 
but the view that an infinite number of instants make 
up an infinitely long time is not true, and therefore 
the conclusion that Achilles will never overtake the 
tortoise does not follow. 

The third argument^ that of the arrow, is very 
interesting. The text has been questioned. Burnet 
accepts the alterations of Zeller, and paraphrases 

* Op. cit. 

Aristotle's words are: "The second is the so-called 
Achilles. It consists in this, that the slower will never be 
overtaken in its course by the quickest, for the pursuer must 
always come first to the point from which the pursued has 
just departed, so that the slower must necessarily be always 
still more or less in advance." Phys., vi. 9. 2393 (R.P. 137). 

I Phys., vi. 9. 2398 (R.P. 138). 


"The arrow in flight is at rest. For, if every- 
thing is at rest when it occupies a space equal to 
itself, and what is in flight at any given moment 
always occupies a space equal to itself, it cannot 

But according to Prantl, the literal translation 
of the unemended text of Aristotle's statement of the 
argument is as follows : " If everything, when it is 
behaving in a uniform manner, is continually either 
moving or at rest, but what is moving is always in 
the now, then the moving arrow is motionless." This 
form of the argument brings out its force more clearly 
than Bumet's paraphrase. 

Here, if not in the first two arguments, the view that 
a finite part of time consists of a finite series of suc- 
cessive instants seems to be assumed ; at any rate 
the plausibility of the argument seems to depend upon 
supposing that there are consecutive instants. 
Throughout an instant, it is said, a moving body is 
where it is : it cannot move during the instant, for 
that would require that the instant should have parts. 
Thus, suppose we consider a period consisting of a 
thousand instants, and suppose the arrow is in flight 
throughout this period. At each of the thousand 
instants, the arrow is where it is, though at the next 
instant it is somewhere else. It is never moving, 
but in some miraculous way the change of position 
has to occur between the instants, that is to say, not 
at any time whatever. This is what M. Bergson calls 
the cinematographic representation of reality. The 
more the difficulty is meditated, the more real it 
becomes. The solution lies in the theory of continuous 
series : we find it hard to avoid supposing that, when 
the arrow is in flight, there is a next position occupied 
at the next moment ; but in fact there is no next 


position and no next moment, and when once 
this is imaginatively realized, the difficulty is seen to 

The fourth and last of Zeno's arguments is x the 
argument of the stadium. 

The argument as stated by Burnet is as follows : 

First Position. Second Position. 

A A 

" Half the time may be equal to double the time. 
Let us suppose three rows of bodies, one of which 
(A) is at rest while the other two (B, C) are moving 
with equal velocity in opposite directions. By the 
time they axe all in the same part of the course, B 
will have passed twice as many of the bodies in C as 
in A. Therefore the time which it takes to pass C 
is twice as long as the time it takes to pass A. But 
the time which B and C take to reach the position of 
A is the same. Therefore double the time is equal 
to the hajf." 

Gayety devoted an interesting article to the inter- 
pretation of this argument. His translation of Aris- 
totle's statement is as follows : 

" The fourth argument is that concerning the two 
rows of bodies, each row being composed of an equal 
number of bodies of equal size, passing each other on 
a race-course as they proceed with equal velocity 
in opposite directions, the one row originally occupying 
the space between the goal and the middle point of 
the course, and the other that between the middle 

' Phys., vi. 9. 2393 (RJP. 139). 
Loc. cit. 


point and the starting-post. This, he thinks, involves 
the conclusion that half a given time is equal to double 
the time. The fallacy of the reasoning lies in the 
assumption that a body occupies an equal time in 
passing with equal velocity a body that is in motion and 
a body of equal size that is at rest, an assumption which 
is false. For instance (so runs the argument), let 
A A . . . be the stationary bodies of equal size, 
BB . . . the bodies, equal in number and in size 
to A A . . ., originally occupying the half of the course 
from the starting-post to the middle of the A's, and 
CC . . . those originally occupying the other half from 
the goal to the middle of the A's, equal in number, size, 
and velocity, to BB . .'. Then three consequences 
follow. First, as the B's and C's pass one another, the 
first B reaches the last C at the same moment at 
which the first C reaches the last B. Secondly, at 
this moment the first C has passed all the A's, whereas 
the first B has passed only half the A's and has conse- 
quently occupied only half the time occupied by the 
first C, since each of the two occupies an equal time 
in passing each A. Thirdly, at the same moment 
all the B's have passed all the C's : for the first C and 
the first B will simultaneously rea^ht the opposite 
ends of the course, since (so says Zeno) the time occupied 
by the first C in passing each of the B's is equal to 
that occupied by it in passing each of the A's, be- 
cause an equal time is occupied by both the first B 
and the first C in passing all the A's. This is the 
argument : but it presupposes the aforesaid fallacious 

This argument is not quite easy to follow, and 
it is only valid as against the assumption that 
a finite time consists of a finite number of instants. 
We may re-state it in different language. Let us 


suppose three drill-sergeants, A, A', and A", standing in 
a row, while the two files of soldiers march past them in 

First Position. Second Position. 

B B' B" B B' B" 

A A' A" A A' A" 

C C' C" C C' C" 

opposite directions. At the first moment which we 
consider, the three men B, B', B" in one row, and the 
three men C, C', C" in the other row, are respectively 
opposite to A, A', and A". At the very next moment, 
each row has moved on, and now B and C" are opposite 
A'. Thus'B and C" are opposite each other. When, 
then, did B pass C' ? It must have been somewhere 
between the two moments which we supposed con- 
secutive, and therefore the two moments cannot really 
have been consecutive. It follows that there must 
be other moments between any two given moments, 
and therefore that there must be an infinite number of 
moments in any given interval of time. 

The above difficulty, that B must have passed C' 
at some time between two consecutive moments, is a 
genuine one, but is not precisely the difficjjlty raised by 
Zeno. What Zeno professes to prove tethat," half of 
a given time is equal to double that time." The most 
intelligible explanation of the argument known to me 
is that of Gaye. 1 Since, however, his explanation is 
not easy to set forth shortly /I will re-state what 
seems to me to be the logical essence of Zeno's conten- 
tion. If we suppose that time consists of a series of 

1 Loc. tit., p. 105. 


consecutive instants, and that motion consists in 
passing through a series of consecutive points, then 
the fastest possible motion is one which, at each instant, 
is at a point consecutive to that at which it was at the 
previous instant. Any slower motion must be one 
which has intervals of rest interspersed, and any 
faster motion must wholly omit some points. All this 
is evident from the fact that we cannot have more than 
one event for each instanj. But now, in the case of 
our A's and B's and C's, B is opposite a fresh A every 
instant, and therefore the number of A's passed gives 
the number of instants since the beginning of the 
motion. But during the motion B has passed twice as 
many C's, and yet cannot have passed more than one 
each instant. Hence the number of instants since the 
motion began is twice the number of A's passed, 
though we previously found it was equal to this number. 
From this result, Zeno's conclusion follows. 

Zeno's arguments, in some form, have afforded 
grounds for almost all the theories of space and time 
and infinity which have been constructed from his 
day to our own. We have seen that all his arguments 
are valid (with certain reasonable hypotheses) on the 
assumption that finite spaces and times consist of a 
finite number of points and instants, and that the 
third and fourth almost certainly in fact proceeded 
on this assumption, while the first and second, which 
were perhaps intended to refute the opppsite assump- 
tion, were in that case fallacious. We may therefore 
escape from his paradoxes either by maintaining 
that, though space and time do consist of points and 
instants, the number of them in any finite interval is 
infinite ; or by denying that space and time consist 
of points and instants at all ; or lastly, by denying 
the reality of space and time altogether. It would 


seem that Zeno himself, as a supporter of Parmenides, 
drew the last of these three possible deductions, at 
any rate in regard to time. In this a very large 
number of philosophers have followed him. Many 
others, like M. Bergson, have preferred to deny that 
space and time consist of points and instants. Either 
of these solutions will meet the difficulties in the form 
in which Zeno raised them. But, as we saw, the diffi- 
culties can also be met if infinite numbers are admis- 
sible. And on grounds which are independent of 
space and time, infinite numbers, and series in which 
no two terms are consecutive, must in any case be 
admitted. Consider, for example, all the fractions 
less than I, arranged in order of magnitude. Between 
any two of them, there are others, for example, the 
arithmetical mean of the two. Thus no two fractions 
are consecutive, and the total number of them is 
infinite. It will be found that much of what Zeno says 
as regards the series of points on a line can be equally 
well applied to the series of fractions. And we cannot 
deny that there are fractions, so that two of the above 
ways of escape are closed to us. It follows that, if 
we are to solve the whole class of difficulties derivable 
from Zeno's by Analogy, we must discover some tenable 
theory of infinite numbers. What, then, are the 
difficulties which, until the last thirty years, led 
philosophers to the belief that infinite numbers are 
impossible ? 

The difficulties of infinity are of two kinds, of which 
the first may be called sham, while the others involve, 
for their solution, a certain amount of new and not 
altogether easy thinking. The sham difficulties are 
those suggested by the etymology, and those suggested 
by confusion of the mathematical infinite with what 
philosophers impertinently call the "true" infinite. 1 


Etymologically, "infinite" should mean "having 
no end." But in fact some infinite series have ends, 
some have not; while some collections are infinite 
without being serial, and can therefore not properly 
be regarded as either endless or having ends. The 
series of instants from any earlier one to any later one 
(both included) is infinite, but has two ends; the 
series of instants from the beginning of time to the 
present moment has one end, but is infinite. Kant, 
in his first antinomy, seems to hold that it is harder 
for the past to be infinite than for the future to be so, 
on the ground that the past is now completed, and 
that nothing infinite can be completed. It is very 
difficult to see how he can have imagined that there 
was any sense in this remark; but it seems most 
probable that he was thinking of the infinite as the 
" unended." It is odd that he did not see that the 
future too has one end at the present, and is precisely 
on a level with the past. His regarding the two as 
different in this respect illustrates just that kind of 
slavery to time which, as we agreed in speaking of 
Pannenides, the true philosopher must learn to leave 
behind him. 

The confusions introduced into the notions of philo- 
sophers by the so-called " true " infinite are curious. 
They see that this notion is not the same as the mathe- 
matical infinite, but they choose to believe that it is 
the notion which the mathematicians are vainly 
trying to reach. They therefore inform the mathe- 
maticians, kindly but firmly, that they are mistaken 
in adhering to the " false " infinite, since plainly the 
"true" infinite is something quite different. The 
reply to this is that what they call the " true " infinite 
is a notion totally irrelevant to the problem of the 
mathematical infinite, to which it has only a fanciful 


and verbal analogy. So remote is it that I do not 
propose to confuse the issue by even mentioning what 
the " true " infinite is. It is the " false " infinite that 
concerns us, and we have to show that the epithet 
" false " is undeserved. 

There are, however, certain genuine difficulties in 
understanding the infinite, certain habits of mind 
derived from the consideration of finite numbers, and 
easily extended to infinite numbers under the mistaken 
notion that they represent logical necessities. For 
example, every number that we are accustomed to, 
except o, has another number immediately before 
it, from which it results by adding i ; but the first 
infinite number does not have this property. The 
numbers before it form an infinite series, containing 
all the ordinary finite numbers, having no maximum, 
no last finite number, after which one little step would 
plunge us into the infinite. , If it is assumed that the 
first infinite number is reached by a succession of 
small steps, it is easy to show that it is self-contra- 
dictory. The first infinite number is, in fact, beyond 
the whole unending series of finite numbers. " But," 
it will be said, "there cannot be anything beyond 
the whole of an unending series." This, we may 
point out, is the very principle upon which Zeno relies 
in the arguments of the race-course and the Achilles. 
Take the race-course : there is the moment when the 
runner still has half his distance to run, then the 
moment when he still has a quarter, then when he still 
has an eighth, and so on in a strictly unending series. 
Beyond the whole of this series is the moment when 
he reaches the goal Thus there certainly can be 
something beyond the whole of an unending series. 
But it remains to show that this fact is only what 
might have been expected. 


The difficulty, like most of the vaguer difficulties 
besetting the mathematical infinite, is derived, I 
think, from the more or less unconscious operation 
of the idea of counting. If you set to work to count 
the terms in an infinite collection, you will never 
have completed your task. Thus, in the case of the 
runner, if half, three-quarters, seven-eighths, and so 
on of the course were marked, and the runner was not 
allowed to pass any of the marks until the umpire 
said "Now," then Zeno's conclusion would be true 
in practice, and he would never reach the goal. 

But it is not essential to the existence of a collection, 
or even to knowledge and reasoning concerning it, 
that we should be able to pass its terms in review one 
by one. This may be seen in the case of finite col- 
lections ; we can speak of " mankind " or " the human 
race," though many of the individuals in this collection 
are not personally known to us. We can do this 
because we know of various characteristics which every 
individual has if he belongs to the collection, and not 
if he does not. And exactly the same happens in the 
case of infinite collections : they may be known by 
their characteristics although their terms cannot be 
enumerated. In this sense, an unending series may 
nevertheless form a whole, and there may be new 
terms beyond the whole of it. 

Some purely arithmetical peculiarities of infinite 
numbers have also caused perplexity. For instance, 
an infinite number is not increased by adding one to 
it, or by doubling it. Such peculiarities have seemed 
to many to contradict logic, but in fact they only 
contradict confirmed mental habits. The whole diffi- 
culty of the subject lies in the necessity of thinking 
in an nnfa-miliflr way, and in realizing that many 
properties which we have thought inherent in number 


are in fact peculiar to finite numbers. If this is 
remembered, the positive theory of infinity, which 
will occupy the next lecture, will not be found so 
difficult as it is to those who ding obstinately to the 
prejudices instilled by the arithmetic which is learnt 
in childhood. 


THE positive theory of infinity, and the general theory 
of number to which it has given rise, are among the 
triumphs of scientific method in philosophy, and are 
therefore specially suitable for illustrating the logical- 
analytic character of that method. The work in this 
subject has been done by mathematicians, and its 
results can be expressed in mathematical symbolism. 
Why, then, it may be said, should the subject be 
regarded as philosophy rather than as mathematics ? 
This raises a difficult question, partly concerned with 
the use of words, but partly also of real importance in 
understanding the function of philosophy. Every 
subject-matter, it would seem, can give rise to philo- 
sophical investigations as well as to the appropriate 
science, the difference between the two treatments 
being in the direction of movement and in the kind of 
truths which it is sought to establish. In the special 
sciences, when they have become fully developed, the 
movement is forward and synthetic, from the simpler 
to the more complex. But in philosophy we follow 
the inverse direction : from the complex and relatively 
concrete we proceed towards the simple and abstract 
by means of analysis, seeking, in the process, to 
eliminate the particularity of the original subject- 


matter, and to confine our attention entirely to the 
logical form of the facts concerned. 

Between philosophy and pure mathematics there is 
a certain affinity, in the fact that both are general 
and a priori. Neither of them asserts propositions 
which, like those of history and geography, depend 
upon the actual concrete facts being just what they 
are. We may illustrate this characteristic by means 
of Leibniz's conception of many possible worlds, of 
which one only is actual. In all the many possible 
worlds, philosophy and mathematics will be the 
same ; the differences will only be in respect of those 
particular facts which are chronicled by the descriptive 
sciences. Any quality, therefore, by which our actual 
world is distinguished from other abstractly possible 
worlds, must be ignored by mathematics and philo- 
sophy alike. Mathematics and philosophy differ, 
however, in their manner of treating the general 
properties in which all possible worlds agree ; for 
while mathematics, starting from comparatively simple 
propositions, seeks to build up more and more complex 
results by deductive synthesis, philosophy, starting 
from data which are common knowledge, seeks to 
purify and generalize them into the simplest statements 
of abstract form that can be obtained from them by 
logical analysis. 

The difference between philosophy and mathematics 
may be illustrated by our present problem, namely 
the nature of number. Both start from certain facts 
about numbers which are evident to inspection. But 
mathematics uses these facts to deduce more and 
more complicated theorems, while philosophy seeks, 
by analysis, to go behind tEese facts to others, simpler, 
more fundamental, and inherently more fitted to form 
the premisses of the science of arithmetic. The 


question, " What is a number ? " is the pre-eminent 
philosophic question in this subject, but it is one which 
the mathematician as such need not ask, provided he 
knows enough of the properties of numbers to enable 
him to deduce his theorems. We, since our object is 
philosophical, must grapple with the philosopher's 
question. The answer to the question, " What is a 
number ? " which we shall reach in this lecture, will 
be found to give also, by implication, the answer to 
the difficulties of infinity which we considered in the 
previous lecture. 

The question " What is a number ? " is one which, 
until quite recent times, was never considered in the 
kind of way that is capable of yielding a precise answer. 
Philosophers were content with some vague dictum 
such as " Number is unity in plurality." A typical 
definition of the kind that contented philosophers is 
the following from Sigwart's Logic ( 66, section 3) : 
" Every number is not merely a plurality, but a plur- 
ality thought as held together and dosed, and to that 
extent as a unity' 9 Now there is in such definitions a 
very elementary blunder, of the same kind that would 
be committed if we said " yellow is a flower " because 
some flowers are yellow. Take, for example, the 
number 3. A single collection of three things might 
conceivably be described as " a plurality thought as 
held together and closed, and to that extent as a 
unity " ; but a collection of three things is not the 
number 3. The number 3 is something which all 
collections of three things have in common, but is not 
itself a collection of three things. The definition, 
therefore, apart from any other defects, has failed to 
reach the necessary degree of abstraction : the number 
3 is something -more abstract than any collection of 
three things. 


Such vague philosophic definitions, however, re- 
mained inoperative because of their very vagueness. 
What most men who thought about numbers really 
had in mind was that numbers are the result of counting. 
" On the consciousness of the law of counting/' says 
Sigwart at the beginning of his discussion of number, 
" rests the possibility of spontaneously prolonging the 
series of numbers ad infinitum." It is this view of 
number as generated by counting which has been the 
chief psychological obstacle to the understanding of 
infinite numbers. Counting, because it is familiar, 
is erroneously supposed to be simple, whereas it is in 
fact a highly complex process, which has no meaning 
unless the numbers reached in counting have some 
significance independent of the process by which they 
axe reached. And infinite numbers cannot be reached 
at all in this way. The mistake is of the same kind 
as if cows were defined as what can be bought from a 
cattle-merchant. To a person who knew several cattle- 
merchants, but had never seen a cow, this might seem 
an admirable definition. But if in his travels he came 
across a herd of wild cows, he would have to declare 
that they were not cows at all, because no cattle- 
merchant could sell them. So infinite numbers were 
declared not to be numbers at all, because they could 
not be reached by counting. 

It will be worth while to consider for a moment what 
counting actually is. We count a set of objects when 
we let our attention pass from one to another, until we 
have attended once to each, saying the names of the 
numbers in order with each successive act of atten- 
tion. The last number named in this process is the 
number of the objects, and therefore counting is a 
method of finding out what the number of the objects 
is. But this operation is really a very complicated 


one, and those who imagine that it is the logical 
source of number show themselves remarkably in- 
capable of analysis. In the first place, when we say 
" one, two, three ..." as we count, we cannot be 
said to be discovering the number of the objects 
counted unless we attach some meaning to the words 
one, two, three. ... A child may learn to know 
these words in order, and to repeat them correctly like 
the letters of the alphabet, without attaching any 
meaning to them. Such a child may count correctly 
from the point of view of a grown-up listener, without 
having any idea of numbers at all. The operation of 
counting, in fact, can only be intelligently performed 
by a person who already has some idea what the 
numbers are ; and from this it follows that counting 
does not give the logical basis of number. 

Again, how do we know that the last number reached 
in the process of counting is the number of the objects 
counted ? This is just one of those facts that are too 
familiar for their significance to be realized; but 
those who wish to be logicians must acquire the habit 
of dwelling upon such facts. There are two proposi- 
tions involved in this fact : first, that the number of 
numbers from i up to any given number is that given 
number for instance, the number of numbers from 
i to 100 is a hundred ; secondly, that if a set of numbers 
can be used as names of a set of objects, each number 
occurring only once, then the number of numbers 
used as names is the same as the number of objects. 
The first of these propositions is capable of an easy 
arithmetical proof so long as finite numbers are con- 
cerned ; but with infinite numbers, after the first, it 
ceases to be true. The second proposition remains 
true, and is in fact, as we shall see, an immediate 
consequence of the definition of number. But owing 



to the falsehood of the first proposition where infinite 
numbers are concerned, counting, even if it were 
practically possible, would not be a valid method of 
discovering the number of terms in an infinite collec- 
tion, and would in fact give different results according 
to the manner in which it was carried out. 

There are two respects in which the infinite numbers 
that are known differ from finite numbers : first, 
infinite numbers have, while finite numbers have not, 
a property which I shall call reflexiveness ; secondly, 
finite numbers have, while infinite numbers have not, 
a property which I shall call inductiveness. Let us 
consider these two properties successively. 

(i) Reflexiveness. A number is said to be reflexive 
when it is not increased by adding i to it. It follows 
at once that any finite number can be added to a 
reflexive number without increasing it. This property 
of infinite numbers was always thought, until recently, 
to be self-contradictory; but through the work of 
Georg Cantor it has come to be recognized that, though 
at first astonishing, it is no more self-contradictory 
than the fact that people at the antipodes do not 
tumble off. In virtue of this property, given any 
infinite collection of objects, any finite number of 
objects can be added or taken away without increasing 
or diminishing the number of the collection. Even 
an infinite number of objects may, under certain 
conditions, be added or taken away without altering 
the number. This may be made clearer by the help 
of some examples. 

Imagine all the natural numbers o, i, 2, 3 ... to 
be written down in a row, and immediately beneath 

O, I, 2, 3 ^ 

i* 2, 3* 4> * r i 


write down the numbers i, 2, 3, 4, . . ., so that i is 
under o, 2 is under i, and so on. Then every number 
in the top row has a number directly under it in the 
bottom row, and no number occurs twice in either 
row. It follows that the number of numbers in the 
two rows must be the same. But all the numbers 
that occur in the bottom row also occur in the top 
row, and one more, namely o ; thus the number of 
terras in the top row is obtained by adding one to 
the number of the bottom row. So long, therefore, 
as it was supposed that a number must be increased 
by adding i to it, this state of things constituted a 
contradiction, and led to the denial that there are 
infinite numbers. 

The following example is even more surprising. 
Write the natural numbers i, 2, 3, 4 ... in the top 
row, and the even numbers 2, 4, 6, 8 ... in the 
bottom row, so that under each number in the top 
row stands its double in the bottom row. Then, as 
before, the number of numbers in the two rows is the 
same, yet the second row results from taking away 
all the odd numbers an infinite collection from the 
top row. This example is given by Leibniz to prove 
that there can be no infinite numbers. He believed 
in infinite collections, but, since he thought that a 
number must always be increased when it is added to 
and diminished when it is subtracted from, he main- 
tained that infinite collections do not have numbers. 
" The number of all numbers," he says, " implies a 
contradiction, which I show thus: To any number 
there is a corresponding number equal to its double. 
Therefore the number of all numbers is not greater 
tfcm the number of even numbers, i.e. the whole is 
not greater thft" its part." x In dealing with this 
Phil. Werke, Gerhardt's edition, vol. i. p. 338. 


argument, we ought to substitute " the number of all 
finite numbers " for " the number of all numbers " ; 
we then obtain exactly the illustration given by our 
two rows, one containing all the finite numbers, the 
other only the even finite numbers. It will be seen 
that Leibniz regards it as self-contradictory to main- 
tain that the whole is not greater than its part. But 
the word " greater " is one which is capable of many 
meanings ; for our purpose, we must substitute the 
less ambiguous phrase " containing a greater number 
of terms." In this sense, it is not self-contradictory 
for whole and part to be equal ; it is the realization 
of this fact which has made the modern theory of 
infinity possible. 

There is an interesting discussion of the reflexiveness 
of infinite wholes in the first of Galileo's Dialogues on 
Motion. I quote from a translation published in 
I730. 1 The personages in the dialogue are Salviati, 
Sagredo, and Simplicius, and they reason as follows : 

" Simp. Here already arises a Doubt which I 
think is not to be resolv'd ; and that is this : Since 
'tis plain that one Line is given greater than another, 
and since both contain infinite Points, we must surely 
necessarily infer, that we have found in the same 
Species something greater than Infinite, since the 
Infinity of Points of the greater Line exceeds the 
Infinity of Points of the lesser. But now, to assign 
an Infinite greater than an Infinite, is what I can't 
possibly conceive. 

i Mathematical Discourses concerning two new sciences 
relating to mechanics and local motion t in four dialogues. By 
Galileo Galilei, Chief Philosopher and Mathematician to the 
Grand Duke of Tuscany. Done into English from the Italian, 
by Tho. Weston, late Master, and now published by John 
Weston, present Master, of the Academy at Greenwich. 
See pp. 46 ff. 


" Salv. These are some of those Difficulties which 
arise from Discourses which our finite Understanding 
makes about Infinites, by ascribing to them Attributes 
which we give to Things finite and terminate, which I 
think most improper, because those Attributes of 
Majority, Minority, and Equality, agree not with 
Infinities, of which we can't say that one is greater 
than, less than, or equal to another. For Proof 
whereof I have something come into my Head, which 
(that I may be the better understood) I will propose 
by way of Interrogatories to Simplicity, who started 
this Difficulty. To begin then : I suppose you know 
which are square Numbers, and which not ? 

" Simp. I know very well that a square Number is 
that which arises from the Multiplication of any 
Number into itself ; thus 4 and 9 are square Numbers, 
that arising from 2, and this from 3, multiplied by 

"Salv. Very well; And you also know, that as 
the Products are calTd Squares, the Factors are calTd 
Roots : And that the other Numbers, which proceed 
not from Numbers multiplied into themselves, are not 
Squares. Whence taking in all Numbers, both Squares 
and Not Squares, if I should say, that the Not Squares 
are more than the Squares, should I not be in the 

" Simp. Most certainly. 

" Salv. If I go on with you then, and ask you, How 
many squar'd Numbers there are? you may truly 
answer, That there are as many as are their proper 
Roots, since every Square has its own Root, and 
every Root its own Square, and since no Square has 
more than one Root, nor any Root more than one 

" Simp. Very true. 


But now, if I should ask how many Roots 
there are, you can't deny but there are as many as 
there are Numbers, since there's no Number but 
what's the Root to some Square. And this being 
granted, we may likewise affirm, that there are as many 
square Numbers, as there are Numbers ; for there are 
as many Squares as there are Roots, and as many 
Roots as Numbers. And yet in the Beginning of this, 
we said, there were many more Numbers than Squares, 
the greater Part of Numbers being not Squares : And 
tho' the Number of Squares decreases in a greater 
proportion, as we go on to bigger Numbers, for count 
to an Hundred you'll find 10 Squares, viz. i, 4, 9, 16, 
25, 36, 49, 64, 81, 100, which is the same as to say the 
loth Part are Squares; in Ten thousand only the 
xooth Part are Squares ; in a Million only the loooth : 
And yet in an infinite Number, if we can but compre- 
hend it, we may say the Squares are as many as all 
the Numbers taken together. 

" Sagr. What must be determin'd then in this 

" Salv. I see no other way, but by saying that all 
Numbers are infinite ; Squares are Infinite, their Roots 
Infinite, and that the Number of Squares is not less 
than the Number of Numbers, nor this less than 
that : and then by concluding that the Attributes 
or Terms of Equality, Majority, and Minority, have 
no Place in Infinites, but are confin'd to terminate 

The way in which the problem is expounded in the 
above discussion is worthy of Galileo, but the solution 
suggested is not the right one. It is actually the case 
that the number of square (finite) numbers is the same 
as the number of (finite) numbers. The fact that, so 
long as we confine ourselves to numbers less than 


some given finite number, the proportion of squares 
tends towards zero as the given finite number increases, 
does not contradict the fact that the number of all 
finite squares is the same as the number of all finite 
numbers. This is only an instance of the fact, now 
familiar to mathematicians, that the limit of a function 
as the variable approaches a given point may not be 
the same as its value when the variable actually reaches 
the given point. But although the infinite numbers 
which Galileo discusses are equal, Cantor has shown 
that what Simplicius could not conceive is true, 
namely that there are an infinite number of different 
infinite numbers, and that the conception of greater 
and less can be perfectly well applied to them. The 
whole of Simplicius's difficulty comes, as is evident, 
from his belief that, if greater and less can be applied, 
a part of an infinite collection must have fewer terms 
than the whole ; and when this is denied, all contra- 
dictions disappear. As regards greater and less 
lengths of lines, which is the problem from which the 
above discussion starts, that involves a meaning of 
greater and less which is not arithmetical The number 
of points is the same in a long line and in a short one, 
being in fact the same as the number of points in all 
space. The greater and less of metrical geometry 
involves the new metrical conception of congruence, 
which cannot be developed out of arithmetical con- 
siderations alone. But this question has not the 
fundamental importance which belongs to the arith- 
metical theory of infinity. 

(2) Non-inductiveness.The second property by 
which infinite numbers are distinguished from finite 
numbers is the property of non-inductiveness. This 
will be best explained by defining the positive property 
of inductiveness which characterizes the finite numbers, 


and which is named after the method of proof known 
as " mathematical induction." 

Let us first consider what is meant by calling a 
property " hereditary " in a given series. Take such 
a property as being named Jones. If a man is named 
Jones, so is his son ; we will therefore call the property 
of being called Jones hereditary with respect to the 
relation of father and son. If a man is called Jones, 
all his descendants in the direct male line are called 
Jones ; this follows from the fact that the property is 
hereditary. Now, instead of the relation of father and 
son, consider the relation of a finite number to its 
immediate successor, that is, the relation which holds 
between o and i, between I and 2, between 2 and 3, 
and so on. If a property of numbers is hereditary 
with respect to this relation, then if it belongs to (say) 
100, it must belong also to all finite numbers greater 
than 100 ; for, being hereditary, it belongs to 101 
because it belongs to 100, and it belongs to 102 because 
it belongs to 101, and so on where the " and so 
on " will take us, sooner or later, to any finite number 
greater than 100. Thus, for example, the property 
of being greater than 99 is hereditary in the series of 
finite numbers ; and generally, a property is hereditary 
in this series when, given any number that possesses 
the property, the next number must always also 
possess it. 

It will be seen that a hereditary property, though 
it must belong to all the finite numbers greater than 
a given number possessing the property, need not 
belong to all the numbers less than this number. For 
example, the hereditary property of being greater 
than 99 belongs to 100 and all greater numbers, but 
not to any smaller number. Similarly, the hereditary 
property of being called Jones belongs to all the 


descendants (in the direct male line) of those who have 
this property, but not to all their ancestors, because 
we reach at last a first Jones, before whom the ancestors 
have no surname. It is obvious, however, that any 
hereditary property possessed by Adam must belong 
to all men; and similarly any hereditary property 
possessed by o must belong to all finite numbers. This 
is the principle of what is called "mathematical 
induction/' It frequently happens, when we wish to 
prove that all finite numbers have some property, 
that we have first to prove that o has the property, 
and then that the property is hereditary, i.e. that, if it 
belongs to a given number, then it belongs to the next 
number. Owing to the fact that such proofs are called 
" inductive," I shall call the properties to which they 
are applicable 1 "inductive" properties. Thus an 
inductive property of numbers is one which is 
hereditary and belongs to o. 

Taking any one of the natural numbers, say 29, it is 
easy to see that it must have all inductive properties. 
For since such properties belong to o and are hereditary, 
they belong to I ; therefore, since they are hereditary, 
they belong to '2, and so on ; by twenty-nine repetitions 
of such arguments we show that they belong to 29. We 
may define the " inductive " numbers as all those thai 
possess att inductive properties ; they will be the same 
as what are called the "natural" numbers, i.e. the 
ordinary finite whole numbers. To all such numbers, 
proofs by mathematical induction can be validly 
applied. They are those numbers, we may loosely say, 
which can be reached from o by successive additions 
of i ; in other words, they are all the numbers that 
can be reached by counting. 

But beyond all these numbers, there are the infinite 
numbers, and infinite numbers do not have all inductive 


properties. Such numbers, therefore, may be called 
non-inductive. All those properties of numbers which 
are proved by an imaginaiy step-by-step process from 
one number to the next are liable to fail when we 
come to infinite numbers. The first of the infinite 
numbers has no immediate predecessor, because there 
is no greatest finite number; thus no succession of 
steps.from one number to the next will ever reach from 
a finite number to an infinite one, and the step-by-step 
method of proof fails. This is another reason for the 
supposed self contradictions of infinite number. Many 
of the most familiar properties of numbers, which 
custom had led people to regard as logically necessary, 
are in fact only demonstrable by the step-by-step 
method, and fail to be true of infinite numbers. But 
so soon as we. realize the necessity of proving such 
properties by mathematical induction, and the strictly 
limited scope of this method of proof, the supposed 
contradictions are seen to contradict, not logic, but 
only our prejudices and mental habits. 

The property of being increased by the addition of 
i i.e. the property of non-reflexiveness may serve to 
illustrate the limitations of mathematical induction. 
It is easy to prove that o is increased by the addition 
of i, and that, if a given number is increased by the 
addition of i, so is the next number, i.e. the number 
obtained by the addition of i. It follows that each 
of the natural numbers is increased by the addition 
of i. This follows generally from the general argument, 
and follows for each particular case by a sufficient 
number of applications of the argument. We first 
prove that o is not equal to i ; then, since the property 
of being increased by i is hereditary, it follows that 
i is not equal to 2 ; hence it follows that 2 is not equal 
to 3 ; if we wish to prove that 30,000 is not equal to 


30,001, we can do so by repeating tM$ reasoning 
30,000 times. But we cannot prove in this way that 
all numbers are increased by the addition of i ; we 
can only prove that this holds of the numbers attain- 
able by successive additions of i starting from o. 
The reflexive numbers, which lie beyond all those 
attainable in this way, are as a matter of fact not 
increased by the addition of i. 

The two properties of reflexiveness and non-induc- 
tiveness, which we have considered as characteristics 
of infinite numbers, have not so far been proved to be 
always found together. It is known that all reflexive 
numbers are non-inductive, but it is not known that 
all non-inductive numbers are reflexive. Fallacious 
proofs of this proposition have been published by many 
writers, including myself, but up to the present no 
valid proof has been discovered. The infinite numbers 
actually known, however, are all reflexive as well as non- 
inductive ; thus, in mathematical practice, if not in 
theory, the two properties are always associated. For 
our purposes, therefore, it will be convenient to ignore 
the bare possibility that there may be non-inductive 
non-reflexive numbers, since all known numbers are 
either inductive or reflexive. 

When infinite numbers are first introduced to 
people, they are apt to refuse the name of numbers to 
them, because their behaviour is so different from that 
of finite numbers that it seems a wilful misuse of terms 
to call them numbers at all. In order to meet this 
feeling, we must now turn to the logical basis of arith- 
metic, and consider the logical definition of numbers. 

The logical definition of numbers, though it seems 
an essential support to the theory of infinite numbers, 
was in fact discovered independently and by a different 
man. The theory of infinite numbers that is to say, 


the arithmetical as opposed to the logical part of the 
theory was discovered by Georg Cantor, and published 
by him in I882-3- 1 The definition of number was 
discovered about the same time by a man whose 
great genius has not received the recognition it deserves 
I mean Gottlob Frege of Jena. His first work, 
Begrifssckrift, published in 1879, contained the very 
important theory of hereditary properties in a series 
to which I alluded in connection with inductiveness. 
His definition of number is conitaned in his second 
work, published in 1884, and entitled Die Grundlagen 
der Arithmetik, eine logisch-mathematische Untersuchung 
fiber den Begri/ der ZaU* It is with this book that 
the logicaJ theory of arithmetic begins, and it will 
repay us to consider Frege's analysis in some detail. 

Frege begins by noting the increased desire for 
logical strictness in mathematical demonstrations 
which distinguishes modern mathematicians from their 
predecessors, and points out that this must lead to a 
critical investigation of the definition of number. He 
proceeds to show the inadequacy of previous philo- 
sophical theories, especially of the " synthetic a priori " 
theory of Kant and the empirical theory of Mill. This 
brings him to the question : What kind of object is it 
that number can properly be ascribed to ? He points 
out that physical things may be regarded as one or 
many : for example, if a tree has a thousand leaves, 
they may be taken altogether as constituting its 

1 In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre 
and in articles in A eta Ma-thematic a, vol. ii. 

* The definition of number contained in this book, and 
elaborated in the Grundgesetee der Arithmetik (vol. i., 1893 ; 
vol. ii., 1903), was rediscovered by me in ignorance of Frege's 
work. I wish to state as emphatically as possible what 
seems still often ignored that his discovery antedated mine 
by eighteen years. 


foliage, which would count as one, not as a thousand ; 
and one pair of boots is the same object as two boots. 
It follows that physical things are not the subjects of 
which number is properly predicated; for when we 
have discovered the proper subjects, the number to 
be ascribed must be unambiguous. This leads to a 
discussion of the very prevalent view that number is 
really something psychological and subjective, a view 
which Frege emphatically rejects. "Number," he 
says, " is as little an object of psychology or an out- 
come of psychical processes as the North Sea. . . . 
The botanist wishes to state something which is just 
as much a fact when he gives the number of petals in 
a flower as when he gives its colour. The one depends 
as little as the other upon our caprice. There is 
therefore a certain similarity between number and 
colour ; but this does not consist in the fact that both 
are sensibly perceptible in external things, but in the 
fact that both are objective " (p. 34). 

" I distinguish the objective," he continues, " from 
the palpable, the spatial, the actual. The earth's 
axis, the centre of mass of the solar system, are objec- 
tive, but I should not call them actual, like the earth 
itself " (p. 35). He concludes that number is neither 
spatial and physical, nor subjective, but non-sensible 
and objective. This conclusion is important, since it 
applies to all the subject-matter of mathematics and 
logic. Most philosophers have thought that the 
physical and the mental between them exhausted the 
world of being. Some have argued that the objects 
of mathematics were obviously not subjective, and 
therefore must be physical and empirical ; others have 
argued that they were obviously not physical, and 
therefore must be subjective and mental Both sides 
were right in what they denied, and wrong in what 


they asserted ; Frege has the merit of accepting both 
denials, and finding a third assertion by recognizing 
the world of logic, which is neither mental nor physical. 

The fact is, as Frege points out, that no number, not 
even i, is applicable to physical things, but only to 
general terms or descriptions, such as " man," " satel- 
lite of the earth," " satellite of Venus." The general 
term "man" is applicable to a certain number of 
objects : there are in the world so and so many men. 
The unity which philosophers rightly feel to be neces- 
sary for the assertion of a number is the unity of the 
general term, and it is the general term which is the 
proper subject of number. And this applies equally 
when there is one object or none which falls under 
the general term. " Satellite of the earth " is a term 
only applicable to one object, namely, the moon. 
But " one " is not a property of the moon itself, which 
may equally well be regarded as many molecules: 
it is a property of the general term " earth's satellite." 
Similarly, o is a property of the general term " satellite 
of Venus," because Venus has no satellite. Here at 
last we have an intelligible theory of the number o. 
This was impossible if numbers applied to physical 
objects, because obviously no physical object could 
have the number o. Thus, in seeking our definition 
of number we have arrived so far at the result that 
numbers are properties of general terms or general 
descriptions, not of physical things or of mental 

Instead of speaking of a general term, such as 
"man," as the subject of which a number can be 
asserted, we may, without making any serious change, 
take the subject as the class or collection of objects 
i.e. " mankind " in the above instance to which the 
general term in question is applicable. Two general 


terms, such as "man" and "featherless biped," 
which are applicable to the same collection of objects, 
will obviously have the same number of instances; 
thus the number depends upon the class, not upon 
the selection of this or that general term to describe 
it, provided several general terms can be found to 
describe the same class. But some general term is 
always necessary in order to describe a class. Even 
when the terms axe enumerated, as " this and that 
and the other," the collection is constituted by the 
general property of being either this, or that, or the 
other, and only so acquires the unity which enables 
us to speak of it as one collection. And in the case 
of an infinite dass, enumeration is impossible, so that 
description by a general characteristic common and 
peculiar to the members of the dass is the only possible 
description. Here, as we see, the theory of number to 
which Frege was led by purdy logical considerations 
becomes of use in showing how infinite classes can be 
amenable to number in spite of being incapable of 

Frege next asks the question: When do two collections 
have the same number of terms ? In ordinary life, 
we decide this question by counting ; but counting, 
as we saw, is impossible in the case of infinite collections, 
and is not logically fundamental with finite collections. 
We want, therefore, a different method of answering 
our question. An illustration may hdp to make the 
method dear. I do not know how many married 
men there are in England, but I do know that the 
number is the same as the number of married women. 
The reason I know this is that the relation of husband 
and wife relates one man to one woman and one woman 
to one man. A relation of this sort is called a one- 
one relation. The relation of father to son is called a 


one-many relation, because a man can have only one 
father but may have many sons; conversely, the 
relation of son to father is called a many-one relation. 
But the relation of husband to wife (in Christian 
countries) is called one-one, because a man cannot 
have more than one wife, or a woman more than one 
husband. Now, whenever there is a one-one relation 
between all the terms of one collection and all the 
terms of another severally, as in the case of English 
husbands and English wives, the number of terms in 
the one collection is the same as the number in the 
other; but when there is not such a relation, the 
number is different. This is the answer to the ques- 
tion : When do two collections have the same number 
of terms ? 

We can now at last answer the question : What is 
meant by the number of terms in a given collection ? 
When there is a one-one relation between all the terms 
of one collection and all the terms of another severally, 
we shall say that the two collections are " similar." 
We have just seen that two similar collections have 
the same number of terms. This leads us to define 
the number of a given collection as the class of all 
collections that are similar to it ; that is to say, we set 
up the following formal definition : 

" The number of terms in a given class " is defined 
as meaning " the class of all classes that are similar 
to the given class." 

This definition, as Frege (expressing it in slightly 
different terms) showed, yields the usual arithmetical 
properties of numbers. It is applicable equally to 
finite and infinite numbers, and it does not require 
the admission of some new and mysterious set of 
metaphysical entities. It shows that it is not physical 
objects, but classes or the general terms by which they 


are defined, of which numbers can be asserted ; and 
it applies to o and i without any of the difficulties 
which other theories find in dealing with these two 
special cases. 

The above definition is sure to produce, at first sight, 
a feeling of oddity, which is liable to cause a certain 
dissatisfaction. It defines the number 2, for instance, 
as the class of all couples, and the number 3 as the 
class of all triads. This does not seem to be what we 
have hitherto been meaning when we spoke of 2 and 3, 
though it would be difficult to say what we had been 
meaning. The answer to a feeling cannot be a logical 
argument, but nevertheless the answer in this case is 
not without importance. In the first place, it will be 
found that when an idea which has grown familiar as 
an unanalysed whole is first resolved accurately into 
its component parts which is what we do when we 
define it there is almost always a feeling of un- 
famiUarity produced by the analysis, which tends to 
cause a protest against the definition. In the second 
place, it may be admitted that the definition, like all 
definitions, is to a certain extent arbitrary. In the 
case of the small finite numbers, such as 2 and 3, it 
would be possible to frame definitions more nearly in 
accordance with our unanalysed feeling of what we 
mean ; but the method of such definitions would lack 
uniformity, and would be found to fail sooner or later 
at latest when we reached infinite numbers. 

In the third place, the real desideratum about such 
a definition as that of number is not that it should 
represent as nearly as possible the ideas of those who 
have not gone through the analysis required in order 
to reach a definition, but that it should give us objects 
having the requisite properties. Numbers, in fact, 
must satisfy the formulae of arithmetic ; any indubit- 



able set of objects fulfilling this requirement may be 
called numbers. So far, the simplest set known to 
fulfil this requirement is the set introduced by the 
above definition. In comparison with this merit, the 
question whether the objects to which the definition 
applies are like or unlike the vague ideas of numbers 
entertained by those who cannot give a definition, is 
one of very little importance. All the important 
requirements are fulfilled by the above definition, and 
the sense of oddity which is at first unavoidable will 
be found to wear off very quickly with the growth 
of familiarity. 

There is, however, a certain logical doctrine which 
may be thought to form an objection to the above 
definition of numbers as classes of classes I mean 
the doctrine that there are no such objects as classes 
at all. It might be thought that this doctrine would 
make havoc of a theory which reduces numbers to 
classes, and of the many other theories in which we 
have made use of classes. This, however, would be a 
mistake : none of these theories are any the worse for 
the doctrine that classes are fictions. What the 
doctrine is, and why it is not destructive, I will try 
briefly to explain. 

On account of certain rather complicated difficulties, 
culminating in definite contradictions, I was led to the 
view that nothing that can be said significantly about 
things, i.e. particulars, can be said significantly (i.e. 
either truly or faJsely) about dasses of things. That 
is to say, if, in any sentence in which a thing is men- 
tioned, you substitute a class for the thing, you no 
longer have a sentence that has any meaning : the 
sentence is no longer either true or false, but a meaning- 
less collection of words. Appearances to the contrary 
can be dispelled by a moment's reflection. For 


example, in the sentence, " Adam is fond of apples," 
you may substitute mankind, and say, " Mankind is 
fond of apples." But obviously you do not mean 
that there is one individual, called " mankind," which 
munches apples : you mean that the separate indi- 
viduals who compose mankind are each severally fond 
of apples. 

Now, if nothing that can be said significantly about 
a thing can be said significantly about a class of things, 
it follows that classes of things cannot have the same 
kind of reality as things have ; for if they had, a dass 
could be substituted for a thing in a proposition 
predicating the kind of reality which would be common 
to both. This view is really consonant to common 
sense. In the third or fourth century B.C. there lived 
a Chinese philosopher named Hui Tzu, who maintained 
that " a bay horse and a dun cow are three ; because 
taken separately they are two, and taken together 
they are one : two and one make three." * The 
author from whom I quote says that Hui Tzu " was 
particularly fond of the quibbles which so delighted 
the sophists or unsound reasoners of ancient Greece," 
and this no doubt represents the judgment of common 
sense upon such arguments. Yet if collections of 
things were things, his contention would be irrefrag- 
able. It is only because the bay horse and the dun 
cow taken together are not a new thing that we can 
escape the conclusion that there are three things 
wherever there are two. 

When it is admitted that classes are not things, the 
question arises : What do we mean by statements 
which are nominally about classes ? Take such a 
statement as, "The dass of people interested in 

i Giles, The Civilisation of China (Home University Library), 
p. 147- 


mathematical logic is not very numerous." Obviously 
this reduces itself to, "Not very many people are 
interested in mathematical logic." For the sake of 
definiteness, let us substitute some particular number, 
say 3, for " very many/' Then our statement is, " Not 
three people are interested in mathematical logic." 
This may be expressed in the form : " If x is interested 
in mathematical logic, and also y is interested, and 
also z is interested, then x is identical with y, or x is 
identical with z 9 or y is identical with z." Here there 
is no longer any reference at all to a " class." In some 
such way, all statements nominally about a class can 
be reduced to statements about what follows from 
the hypothesis of anything's having the defining 
property of the class. All that is wanted, therefore, 
in order to render the verbal use of classes legitimate, 
is a uniform method of interpreting propositions in 
which such a use occurs, so as to obtain propositions 
in which there is no longer any such use. The defini- 
tion of such a method is a technical matter, which 
Dr. Whitehead and I have dealt with elsewhere, and 
which we need not enter into on this occasion. 1 

If the theory that classes are merely symbolic is 
accepted, it follows that numbers are not actual 
entities, but that propositions in which numbers 
verbally occur have not really any constituents corre- 
sponding to numbers, but only a certain logical form 
which is not a part of propositions having this form. 
This is in fact the case with all the apparent objects 
of logic and mathematics. Such words as or, not, 
if, there is, identity, greater, plus, nothing, everything, 
function, and so on, are not names of definite objects, 
like " John " or " Jones," but are words which require 

* Of. Principia Mathematics, 20, and Introduction, 
chapter iii. 


a context in order to have meaning. All of them are 
formal, that is to say, their occurrence indicates a 
certain form of proposition, not a certain constituent. 
" Logical constants," in short, are not entities ; the 
words expressing them are not names, and cannot 
significantly be made into logical subjects except 
when it is the words themselves, as opposed to their 
meanings, that are being discussed. 1 This fact has a 
very important bearing on all logic and philosophy, 
since it shows how they differ from the special sciences. 
But the questions raised are so large and so difficult 
that it is impossible to pursue them further on this 

1 See Tractates Logico-Phihsophicus, by Lad-wig Wittgen- 
stein (Kegan Paul, 1922). 



THE nature of philosophic analysis, as illustrated in 
our previous lectures, can now be stated in general 
terms. We start from a body of common knowledge, 
which constitutes our data. On examination, the 
data are found to be complex, rather vague, and 
largely interdependent logically. By analysis we 
reduce them to propositions which are as nearly as 
possible simple and precise, and we arrange them in 
deductive chains, in which a certain number of initial 
propositions form a logical guarantee for all the rest. 
These initial propositions are premisses for the body 
of knowledge in question. Premisses are thus quite 
different from data they are simpler, more precise, 
and less infected with logical redundancy. If the 
work of analysis has been performed completely, they 
will be wholly free from logical redundancy, wholly 
precise, and as simple as is logically compatible with 
their leading to the given body of knowledge. The 
discovery of these premisses belongs to philosophy ; 
but the work of deducing the body of common know- 
ledge from them belongs to mathematics, if " mathe- 
matics " is interpreted in a somewhat liberal sense. 

But besides the logical analysis of the common 
knowledge which forms our data, there is the considera- 


tion of its degree of certainty. When we have arrived 
at its premisses, we may find that some of them seem 
open to doubt, and we may find further that this 
doubt extends to those of our original data which 
depend upon these doubtful premisses. In our third 
lecture, for example, we saw that the part of physics 
which depends upon testimony, and thus upon the 
existence of other minds than our own, does not seem 
so certain as the part which depends exclusively upon 
our own sense-data and the laws of logic. Similarly, 
it used to be felt that the parts of geometry which 
depend upon the axiom of parallels have less certainty 
than the parts which are independent of this premiss. 
We may say, generally, that what commonly passes 
as knowledge is not all equally certain, and that, 
when analysis into premisses has been effected, the 
degree of certainty of any consequence of the premisses 
will depend upon that of the most doubtful premiss 
employed in proving this consequence. Thus analysis 
into premisses serves not only a logical purpose, but 
also the purpose of facilitating an estimate as to the 
degree of certainty to be attached to this or that 
derivative belief. In view of the fallibility of all 
human beliefs, this service seems at least as important 
as the purely logical services rendered by philosophical 

In the present lecture, I wish to apply the analytic 
method to the notion of " cause, 11 and to illustrate 
the discussion by applying it to the problem of free 
will. For this purpose I shall inquire: I, what is 
meant by a causal law ; II, what is the evidence that 
causal laws have held hitherto ; III, what is the 
evidence that they will continue to hold in the future ; 
IV, how the causality which is used in science differs 
from that of common sense and traditional philosophy ; 


V, what new light is thrown on the question of free 
will by our analysis of the notion of " cause." 

I. By a " causal law " I mean any general proposi- 
tion in virtue of which it is possible to infer the exist- 
ence of one thing or event from the existence of another 
or of a number of others. If you hear thunder without 
having seen lightning, you infer that there neverthe- 
less was a flash, because of the general proposition, 
"All thunder is preceded by lightning." When 
Robinson Crusoe sees a footprint, he infers a human 
being, and he might justify his inference by the general 
proposition, " All marks in the ground shaped like a 
human foot are subsequent to a human being's standing 
where the marks are." When we see the sun set, we 
expect that it will rise again the next day. When we 
hear a man speaking, we infer that he has certain 
thoughts. All these inferences are due to causal 

A causal law, we said, allows us to infer the existence 
of one thing (or event) from the existence of one or more 
others. The word " thing " here is to be understood 
as only applying to particulars, i.e. as excluding such 
logical objects as numbers or classes or abstract 
properties and relations, and including sense-data, 
with whatever is logically of the same type as sense- 
data. 1 In so far as a causal law is directly verifiable, 
the thing inferred and the thing from which it is 
inferred must both be data, though they need not 
both be data at the same time. In fact, a causal law 
which is being used to extend our knowledge of exist- 
ence must be applied to what, at the moment, is not a 
datum; it is in the possibility of such application 

* Thus we are not using "thing " here in the sense of a class 
of correlated "aspects/' as we did in Lecture III. Each 
" aspect " will count separately in stating causal laws. 


that the practical utility of a causal law consists. 
The important point, for our present purpose, however, 
is that what is inferred is a " thing," a " particular," 
an object having the kind of reality that belongs to 
objects of sense, not an abstract object such as virtue 
or the square root of two. 

But we cannot become acquainted with a particular 
except by its being actually given. Hence the par- 
ticular inferred by a causal iaw must be only described 
with more or less exactness ; it cannot be named until 
the inference is verified. Moreover, since the causal 
law is general, and capable of applying to many cases, 
the given particular from which we infer must allow 
the inference in virtue of some general characteristic, 
not in virtue of its being just the particular that it is. 
This is obvious in all our previous instances : we infer 
the unperceived lightning from the thunder, not in 
virtue of any peculiarity of the thunder, but in virtue 
of its resemblance to other daps of thunder. Thus a 
causal law must state that the existence of a thing of 
a certain sort (or of a number of things of a number of 
assigned sorts) implies the existence of another thing 
having a relation to the first which remains invariable 
so long as the first is of the kind in question. 

It is to be observed that what is constant in a causal 
law is not the object or objects given, nor yet the 
object inferred, both of which may vary within wide 
limits, but the relation between what is given and 
what is inferred. The principle, " same cause, same 
effect," which is sometimes said to be the principle of 
causality, is much narrower in its scope than the 
principle which really occurs in science; indeed, if 
strictly interpreted, it has no scope at all, since the 
" same " cause never recurs exactly. We shall return 
to this point at a later stage of the discussion. 


The particular which is inferred may be uniquely 
determined by the causal law, or may be only described 
in such general terms that many different particulars 
might satisfy the description. This depends upon 
whether the constant relation affirmed by the causal 
law is one which only one term can have to the data, 
or one which many terms may have. If many terms 
may have the relation in question, science will not be 
satisfied until it has found some more stringent law, 
which will enable us to determine the inferred things 

Since all known things are in time, a causal law 
must take account of temporal relations. It will be 
part of the causal law to state a relation of succession 
or coexistence between the thing given and the thing 
inferred. When we hear thunder and infer that there 
was lightning, the law states that the thing inferred 
is earlier than the thing given. Conversely, when we 
see lightning and wait expectantly for the thunder, 
the law states that the thing given is earlier than the 
thing inferred. When we infer a man's thoughts 
from his words, the law states that the two are (at 
least approximately) simultaneous. 

If a causal law is to achieve the precision at which 
science aims, it must not be content with a vague 
earlier or later, but must state how much earlier or 
how much later. That is to say, the time-relation 
between the thing given and the thing inferred ought 
to be capable of exact statement; and usually the 
inference to be drawn is different according to the 
length and direction of the interval. " A quarter of 
an hour ago this man was alive ; an hour hence he 
will be cold." Such a statement involves two causal 
laws, one inferring from a datum something which 
existed a quarter of an hour ago, the other inferring 


from the same datum something which will exist an 
hour hence. 

Often a causal law involved not one datum, but 
many, which need not be all simultaneous with each 
other, though their time-relations must be given. 
The general scheme of a causal law will be as 
follows : 

" Whenever things occur in certain relations to each 
other (among which their time-relations must be 
included), then a thing having a fixed relation to these 
things will occur at a date fixed relatively to their 

The things given will not, in practice, be things that 
only exist for an instant, for such things, if there are 
any, can never be data. The things given will each 
occupy some finite time. They may be not static 
things, but processes, especially motions. We have 
considered in an earlier lecture the sense in which a 
motion may be a datum, and need not now recur to 
this topic. 

It is not essential to a causal law that the object 
inferred should be later than some or all of the data. 
It may equally well be earlier or at the same time. 
The only thing essential is that the law should be such 
as to enable us to infer the existence of an object which 
we can more or less accurately describe in terms of 
the data. 

II. I come now to our second question, namely : 
What is the nature of the evidence that causal laws 
have held hitherto, at least in the observed portions 
of the past ? This question must not be confused with 
the further question : Does this evidence warrant us 
in assuming the truth of causal laws in the future 
and in unobserved portions of the past? For the 
present, I am only asking what are the grounds which 


lead to a belief in causal laws, not whether these 
grounds axe adequate to support the belief in universal 

The first step is the discovery of approximate un- 
analysed uniformities of sequence or coexistence. 
After lightning comes thunder, after a blow received 
comes pain, after approaching a fire comes warmth ; 
again, there are uniformities of coexistence, for ex- 
ample between touch and sight, between certain 
sensations in the throat and the sound of one's own 
voice, and so on. Every such uniformity of sequence 
or coexistence, after it has been experienced a certain 
number of times, is followed by an expectation that it 
will be repeated on future occasions, i.e. that where one 
of the correlated events is found, the other will be 
found also. The connection of experienced past 
uniformity with expectation as to the future is just 
one of those uniformities of sequence which we have 
observed to be true hitherto. This affords a psycho- 
logical account of what may be called the animal belief 
in causation, because it is somet-hing which can be 
observed in horses and dogs, and is rather a habit of 
acting than a real belief. So far, we have merely 
repeated Hume, who carried the discussion of cause up 
to this point, but did not, apparently, perceive how 
much remained to be said. 

Is there, in fact, any characteristic, such as might 
be called causality or uniformity, which is found to 
hold throughout the observed past ? And if so, how 
is it to be stated ? 

The particular uniformities which we mentioned 
before, such as lightning being followed by thunder, 
are not found to be free from exceptions. We some- 
times see lightning without hearing thunder ; and 
although, in such a case, we suppose that thunder 


might have been heard if we had been nearer to the 
lightning, that is a supposition based on theory, and 
therefore incapable of being invoked to support the 
theory. What does seem, however, to be shown by 
scientific experience is this : that where an observed 
uniformity fails, some wider uniformity can be found, 
embracing more circumstances, and subsuming both 
the successes and the failures of the previous uniform- 
ity. Unsupported bodies in air fall, unless they are 
balloons or aeroplanes ; but the principles of mechanics 
give uniformities which apply to balloons and aero- 
planes just as accurately as to bodies that fall. There 
is much that is hypothetical and more or less artificial 
in the uniformities affirmed by mechanics, because, 
when they cannot otherwise be made applicable, 
unobserved bodies are inferred in order to account for 
observed peculiarities. Still, it is an empirical fact 
that it is possible to preserve the laws by assuming 
such bodies, and that they never have to be assumed 
in drcumstances in which they ought to be observable. 
Thus the empirical verification of mechanical laws 
may be admitted, although we must also admit that 
it is less complete and triumphant than is sometimes 

Assuming now, what must be admitted to be doubt- 
ful, that the whole of the past has proceeded according 
to invariable laws, what can we say as to the nature 
of these laws ? They will not be of the simple type 
which asserts that the same cause always produces 
the same effect. We may take the law of gravitation 
as a sample of the kind of law that appears to be 
verified without exception. In order to state this 
law in a form which observation can confirm, we will 
confine it to the solar system. It then states that the 
motions of planets and their satellites have at every 


instant an acceleration compounded of accelerations 
towards all the other bodies in the solar system, 
proportional to the matters of these bodies and 
inversely proportional to the squares of their distances. 
In virtue of this law, given the state of the solar system 
throughout any finite time, however short, its state at 
all earlier and later times is determinate except in so 
far as other forces than gravitation or other bodies 
than those in the solar system have to be taken into 
consideration. But other forces, so far as science 
can discover, appear to be equally regular, and equally 
capable of being summed up in single causal laws. If 
the mechanical account of matter were complete, the 
whole physical history of the universe, past and 
future, could be inferred from a sufficient number of 
data concerning an assigned finite time, however 

In the mental world, the evidence for the universality 
of causal laws is less complete than in the physical 
world. Psychology cannot boast of any triumph 
comparable to gravitational astronomy. Nevertheless- 
the evidence is not very greatly less than in the physical 
world. The crude and approximate causal laws from 
which science starts are just as easy to discover in the 
mental sphere as in the physical. In the world of 
sense, there axe to begin with the correlations of sight 
and touch and so on, and the facts which lead us to 
connect various kinds of sensations with eyes, ears, 
nose, tongue, etc. Then there are such facts as that 
our body moves in answer to our volitions. Excep- 
tions exist, but are capable of being explained as 
easily as the exceptions to the rule that unsupported 
bodies in air fall. There is, in fact, just such a degree 
of evidence for causal laws in psychology as will 
warrant the psychologist in assuming them as a matter 


of course, though not such a degree as will suffice to 
remove all doubt from the mind of a sceptical inquirer. 
It should be observed that causal laws in which the 
given term is mental and the inferred term physical, 
or vice versa, are at least as easy to discover as causal 
laws in which both terms are mental. 

It will be noticed that, although we have spoken of 

causal laws, we have not hitherto introduced the 

word " cause." At this stage, it will be well to say a 

few words on legitimate and illegitimate uses of this 

word. The word " cause," in the scientific account of 

the world, belongs only to the early stages, in which 

small preliminary, approximate generalizations are 

being ascertained with a view to subsequent larger 

and more invariable laws. We may say "Arsenic 

causes death," so long as we are ignorant of the precise 

process by which the result is brought about. But in 

a sufficiently advanced science, the word "cause" 

will not occur in any statement of invariable laws. 

There is, however, a somewhat rough and loose use 

of the word " cause " which may be preserved. The 

approximate uniformities which lead to its pre-scientific 

employment may turn out to be true in all but very 

rare and exceptional circumstances, perhaps in all 

circumstances that actually occur. In such cases, it 

is convenient to be able to speak of the antecedent 

event as the " cause " and the subsequent event as 

the " effect." In this sense, provided it is realized 

that the sequence is not necessary and may have 

exceptions, it is still possible to employ the words 

" cause " and " effect." It is in this sense, and in this 

sense only, that we shall intend the words when we 

speak of one particular event "causing" another 

particular event, as we must sometimes do if we are 

to avoid intolerable drcumlocution. 


III. We come now to our third question, namely : 
What reason can be given for believing that causal 
laws will hold in future, or that they have held in 
unobserved portions of the past ? 

What we have said so far is that there have been 
hitherto certain observed causal laws, and that all the 
empirical evidence we possess is compatible with the 
view that everything, both mental and physical, so 
far as our observation has extended, has happened in 
accordance with causal laws. The law of universal 
causation, suggested by these facts, may be enunciated 
as follows : 

" There are such invariable relations between differ- 
ent events at the same or different times that, given 
the state of the whole universe throughout any finite 
time, however short, every previous and subsequent 
event can theoretically be determined as a function 
of the given events during that time." 

Have we any reason to believe this universal law ? 
Or, to ask a more modest question, have we any 
reason to believe that a particular causal law, such as 
the law of gravitation, will continue to hold in the 
future ? 

Among observed causal laws is this, that observation 
of uniformities is followed by expectation of their 
recurrence. A horse who has been driven always 
along a certain road expects to be driven along that 
road again ; a dog who is always fed at a certain hour 
expects food at that hour and not at any other. Such 
expectations, as Hume pointed out, explain only too 
well the common-sense belief in uniformities of se- 
quence, but they afford absolutely no logical ground 
for beliefs as to the future, not even for the belief 
that we shall continue to expect the continuation of 
experienced uniformities, for that is precisely one of 


those causal laws for which a ground has to be sought. 
If Hume's account of causation is the last word, 
we have not only no reason to suppose that the 
sun will rise to-morrow, but no reason to suppose 
that five minutes hence we shall still expect it to rise 

It may, of course, be said that all inferences as to 
the future are in fact invalid, and I do not see how 
such a view could be disproved. But, while admitting 
the legitimacy of such a view, we may nevertheless 
inquire : If inferences as to the future are valid, what 
principle must be involved in making them ? 

The principle involved is the principle of induction, 1 
which, if it is true, must be an a priori logical law, not 
capable of being proved or disproved by experience. 
It is a difficult question how this principle ought to be 
formulated ; but if it is to warrant the inferences 
which we wish to make by its means, it must lead to 
the following proposition : " If, in a great number of 
instances, a tiling of a certain kind is associated in a 
certain way with a thing of a certain other kind, it is 
probable that a thing of the one kind is always similarly 
associated with a thing of the other kind ; and as the 
number of instances increases, the probability ap- 
proaches indefinitely near to certainty." It may 
well be questioned whether this proposition is true ; 
but if we admit it, we can infer that any characteristic 
of the whole of the observed past is likely to apply 
to the future and to the unobserved past. This 
proposition, therefore, if it is true, will warrant the 
inference that causal laws probably hold at all times, 
future as well as past; but without this principle, 
the observed cases of the truth of causal laws afford 

xOn this subject, see Keynes's Treatise on Probability 
t 1921). 



no presumption as to the unobserved cases, and 
therefore the existence of a thing not directly observed 
can never be validly inferred. 

It is thus the principle of induction, rather than the 
law of causality, which is at the bottom of all inferences 
as to the existence of things not immediately given. 
With the principle of induction, all that is wanted for 
such inferences can be proved; without it, all such 
inferences are invalid. This principle has not received 
the attention which its great importance deserves. 
Those who were interested in deductive logic naturally 
enough ignored it, while those who emphasized the 
scope of induction wished to maintain that all logic 
is empirical, and therefore could not be expected to 
realize that induction itself, their own darling, required 
a logical principle which obviously could not be proved 
inductively, and must therefore be a priori if it could 
be known at all. 

The view that the law of causality itself is a priori 
cannot, I think, be maintained by anyone who rftfl.1i7.e3 
what a complicated principle it is. In the form 
which states that " every event has a cause " it looks 
simple ; but on examination, " cause " is merged in 
" causal law," and the definition of a " causal law " is 
found to be far from simple. There must necessarily 
be some a priori principle involved in inference from 
the existence of one thing to that of another, if such 
inference is ever valid ; but it would appear from the 
above analysis that the principle in question is induc- 
tion, not causality. Whether inferences from past to 
future are valid depends wholly if our discussion has 
been sound, upon the inductive principle : if it is true, 
such inferences are valid, and if it is false, they are 

IV. I come now to the question how the conception 


of causal laws which we have arrived at is related to 
the traditional conception of cause as it occurs in 
philosophy and common sense. 

Historically, the notion of cause has been bound 
up with that of human volition. The typical cause 
would be the fiat of a king. The cause is supposed 
to be " active," the effect " passive." From this it 
is easy to pass on to the suggestion that a " true " 
cause must contain some prevision of the effect ; 
hence the effect becomes the " end " at which the 
cause aims, and teleology replaces causation in the 
explanation of nature. But all such ideas, as applied 
to physics, are mere anthropomorphic superstitions. 
It is as a reaction against these errors that Mach and 
others have urged a purely " descriptive " view of 
physics : physics, they say, does not aim at telling us 
" why " things happen, but only " how " they happen. 
And if the question " why ? " means anything more 
than the search for a general law according to which 
a phenomenon occurs, then it is certainly the case 
that this question cannot be answered in physics and 
ought not to be asked. In this sense, the descriptive 
view is indubitably in the right. But in using causal 
laws to support inferences from the observed to the 
unobserved, physics ceases to be pwdy descriptive, 
and it is these laws which give the scientifically useful 
part of the traditional notion of " cause." There is 
therefore something to preserve in this notion, though 
it is a very tiny part of what is commonly assumed in 
orthodox metaphysics. 

In order to understand the difference between the 
kind of cause which science uses and the kind which 
we naturally imagine, it is necessary to shut out, by 
an effort, everything that differentiates between past 
and future. This is an extraordinarily difficult thing 


to do, because our mental life is so intimately bound 
up with difference Not only do memory and hope 
make a difference in our feelings as regards past and 
future, but almost our whole vocabulary is filled with 
the idea of activity, of things done now for the sake 
of their future effects. All transitive verbs involve 
the notion of cause as activity, and would have to be 
replaced by some cumbrous periphrasis before this 
notion could be eliminated. 

Consider such a statement as, " Brutus killed Caesar." 
On another occasion, Brutus and Caesar might engage 
our attention, but for the present it is the killing that 
we have to study. We may say that to kill a person 
is to cause his death intentionally. This means that 
desire for a person's death causes a certain act, because 
it is believed that that act will cause the person's 
death ; or more accurately, the desire and the belief 
jointly cause the act. Brutus desires that Caesar 
should be dead, and believes that he will be dead if 
he is stabbed ; Brutus therefore stabs him, and the 
stab causes Caesar's death, as Brutus expected it would. 
Every act which realizes a purpose involves two causal 
steps in this way : C is desired, and it is believed 
(truly if the purpose is achieved) that B will cause C ; 
the desire and the belief together cause B, which in 
turn causes C. Thus we have first A, which is a desire 
for C and a belief that B (an act) will cause C ; then 
we have B, the act caused by A, and believed to be 
a cause of C ; then, if the belief was correct, we have 
C, caused by B, and if the belief was incorrect we have 
disappointment. Regarded purely scientifically, this 
series A, B, C may equally well be considered hi the 
inverse order, as they would be at a coroner's inquest. 
But from the point of view of Brutus, the desire, which 
comes at the beginning, is what makes the whole 


series interesting. We feel that if his desires had been 
different, the effects which he in fact produced would 
not have occurred. This is true, and gives him a 
sense of power and freedom. It is equally true that 
if the effects had not occurred, his desires would have 
been different, since being what they were the effects 
did occur. Thus the desires are determined by their 
consequences just as much as the consequences by the 
desires ; but as we cannot (in general) know in advance 
the consequences of our desires without knowing our 
desires, this form of inference is uninteresting as applied 
to our own acts, though quite vital as applied to those 
of others. 

A cause, considered scientifically, has none of that 
analogy with volition which makes us imagine that the 
effect is compelled by it. A cause is an event or group 
of events of some known general character, and having 
a known relation to some other event, called the 
effect ; the relation being of such a kind that only one 
event, or at any rate only one well-defined sort of 
event, can have the relation to a given cause. It is 
customary only to give the name "effect" to an 
event which is later than the cause, but there is no 
kind of reason for this restriction. We shall do better 
to allow the effect to be before the cause or simultane- 
ous with it, because nothing of any scientific importance 
depends upon its being after the cause. 

If the inference from cause to effect is to be indubit- 
able, it seems that the cause can hardly stop short of 
the whole universe. So long as anything is left out, 
something may be left out which alters the expected 
result. But for practical and scientific purposes, 
phenomena can be collected into groups which are 
causally self-contained, or nearly so. In the common 
notion of causation, the cause is a single event we say 


the lightning causes the thunder, and so on. But it is 
difficult to know what we mean by a single event ; and 
it generally appears that, in order to have anything 
approaching certainty concerning the effect, it is 
necessary to include many more circumstances in the 
cause than unscientific common sense would suppose. 
But often a probable causal connection, where the 
cause is fairly simple, is of more practical importance 
than a more indubitable connection in which the 
cause is so complex as to be hard to ascertain. 

To sum up: the strict, certain, universal law of 
causation which philosophers advocate is an ideal, 
possibly true, but not known to be true in virtue of 
any available evidence. What is actually known, as 
a matter of empirical science, is that certain constant 
relations are observed to hold between the members 
of a group of events at certain times, and that when 
such relations fail, as they sometimes do, it is usually 
possible to discover a new, more constant relation 
by enlarging the group. Any such constant relation 
between events of specified kinds with given intervals 
of time between them is a "causal law." But all 
causal laws are liable to exceptions, if the cause is 
less than the whole state of the universe ; we believe, 
on the basis of a good deal of experience, that such 
exceptions can be dealt with by enlarging the group 
we caJl the cause, but this belief, wherever it is still 
unverified, ought not to be regarded as certain, but 
only as suggesting a direction for further inquiry. 

A very common causal group consists of volitions 
and the consequent bodily acts, though exceptions 
arise (for example) through sudden paralysis. Another 
very frequent connection (though here the exceptions 
are much more numerous) is between a bodily act and 
the realization of the purpose which led to the act. 


These connections are patent, whereas the causes of 
desires are more obscure. Thus it is natural to begin 
causal series with desires, to suppose that all causes 
are analogous to desires, and that desires themselves 
arise spontaneously. Such a view, however, is not 
one which any serious psychologist would maintain. 
But this brings us to the question of the application 
of our analysis of cause to the problem of free will 

V. The problem of free will is so intimately bound 
up with the analysis of causation that, old as it is, we 
need not despair of obtaining new light on it by the 
help of new views on the notion of cause. The free-will 
problem has, at one time or another, stirred men's 
passions profoundly, and the fear that the will might 
not be free has been to some men a source of great 
unhappiness. I believe that, under the influence of a 
cool analysis, the doubtful questions involved will be 
found to have no such emotional importance as is 
sometimes thought, since the disagreeable conse- 
quences supposed to flow from a denial of free will do 
not flow from this denial in any form in which there 
is reason to make it. It is not, however, on this 
account chiefly that I wish to discuss this problem, 
but rather because it affords a good example of the 
clarifying effect of analysis and of the interminable 
controversies which may result from its neglect. 

Let us first try to discover what it is we really 
desire when we desire free will Some of our reasons 
for desiring free will are profound, some trivial. To 
begin with the former : we do not wish to fed ourselves 
in the hands of fate, so that, however much we may 
desire to will one thing, we may nevertheless be com- 
pelled by an outside force to will another. We do 
not wish to think that, however much we may desire 
ty> act well, heredity and surroundings may f ores us 


into acting ill. We wish to feel that, in cases of 
doubt, our choice is momentous and lies within our 
power. Besides these desires, which are worthy of 
all respect, we have, however, others not so respectable, 
which equally make us desire free will. We do not 
like to think that other people, if they knew enough, 
could predict our actions, though we know that we 
can cf ten predict those of other people, especially if 
they are elderly. Much as we esteem the old gentle- 
man who is our neighbour in the country, we know 
that when grouse are mentioned he will tell the story 
of the grouse in the gun-room. But we ourselves are 
not so mechanical : we never tell an anecdote to the 
same person twice, or even once unless he is sure to 
enjoy it ; although we once met (say) Bismarck, we 
are quite capable of hearing him mentioned without 
relating the occasion when we met him. In this 
sense, everybody thinks that he himself has free will, 
though he knows that no one else has. The desire 
for .this kind of free will seems to be no better than a 
form of vanity. I do not believe that this desire can 
be gratified with any certainty ; but the other, more 
respectable desires are, I believe, not inconsistent 
with any tenable form of determinism. 

We have thus two questions to consider : (i) Are 
human actions theoretically predictable from a suffi- 
cient number of antecedents ? (2) Are human actions 
subject to an external compulsion ? The two ques- 
tions, as I shall try to show, are entirely distinct, and 
we may answer the first in the affirmative without 
therefore being forced to give an affirmative answer 
to the second. 

(i) Are human actions theoretically predictable from 
a sufficient number of antecedents? Let us first en- 
deavour to give precision to this question. We may 


state the question thus : Is there some constant relation 
between an act and a certain number of earlier events, 
such that, when the earlier events are given, only one 
act, or at most only acts with some well-marked 
character, can have this relation to the earlier events ? 
If this is the case, then, as soon as the earlier events 
are known, it is theoretically possible to predict either 
the precise act, or at least the character necessary to 
its fulfilling the constant relation. 

To this question, a negative answer has been given 
by Bergson, in a form which calls in question the 
general applicability of the law of causation. He 
maintains that every event, and more particularly 
every mental event, embodies so much of the past 
that it could not possibly have occurred at any earlier 
time, and is therefore necessarily quite different from 
all previous and subsequent events. If, for example, 
I read a certain poem many times, my experience on 
each occasion is modified by the previous readings, 
and my emotions are never repeated exactly. The 
principle of causation, according to him, asserts that 
the same cause, if repeated, will produce the same 
effect. But owing to memory, he contends, this 
principle does not apply to mental events. What is 
apparently the same cause, if repeated, is modified 
by the mere fact of repetition, and cannot produce 
the same effect. He infers that every mental event 
is a genuine novelty, not predictable from the past, 
because the past contains nothing exactly like it by 
which we could imagine it. And on this ground he 
regards the freedom of the will as unassailable. 

Bergson's contention has undoubtedly a great deal 
of truth, and I have no wish to deny its importance. 
But I do not think its consequences are quite what 
he believes them to be. It is not necessary for the 


detemunist to maintain that he can foresee the whole 
particularity of the act which will be performed. If 
he could foresee that A was going to murder B, his 
foresight would not be invalidated by the fact that he 
could not know all the infinite complexity of A's state 
of mind in committing the murder, nor whether the 
murder was to be performed with a knife or with a 
revolver. .If the kind of act which will be performed 
can be foreseen within narrow limits, it is of little 
practical interest that there are fine shades which 
cannot be foreseen. No doubt every time the story 
of the grouse in the gun-room is told, there will be 
slight differences due to increasing habitualness, but 
they do not invalidate the prediction that the story 
will be told. And there is nothing in Bergson's 
argument to show that we can never predict what 
kind of act will be performed. 

Again, his statement of the law of causation is 
inadequate. The law does not state merely that, if 
the same cause is repeated, the same effect will result. 
It states rather that there is a constant relation between 
causes of certain kinds and effects of certain kinds. 
For example, if a body falls freely, there is a constant 
relation between the height through which it falls and 
the time it takes in falling. It is not necessary to have 
a body fall through the same height which has been 
previously observed, in order to be able to foretell the 
length of time occupied in falling. If this were 
necessary, no prediction would be possible, since it 
would be impossible to make the height exactly the 
same on two occasions. Similarly, the attraction 
which the sun will exert on the earth is not only known 
at distances for which it has been observed, but at all 
distances, because it is known to vary as the inverse 
square of the distance, III f act, what i foun4 to b? 


repeated is always the relation of cause and effect, not 
the cause itself ; all that is necessary as regards the 
cause is that it should be of the same kind (in the 
relevant respect) as earlier causes whose effects have 
been observed. 

Another respect in which Bergson's statement of 
causation is inadequate is in its assumption that the 
cause must be one event, whereas it may be two or 
more events, or even some continuous process. The 
substantive question at issue is whether mental events 
are determined by the past. Now in such a case as 
the repeated reading of a poem, it is obvious that our 
feelings in reading the poem are most emphatically 
dependent upon the past, but not upon one single 
event in the past. All our previous readings of the 
poem must be included in the cause. But we easily 
perceive a certain law according to which the effect 
varies as the previous readings increase in number, 
and in fact Bergson himself tacitly assumes such a 
law. We decide at last not to read the poem again, 
because we know that this time the effect would be 
boredom. We may not know all the niceties and 
shades of the boredom we should fed, but we know 
enough to guide our decision, and the prophecy of 
boredom is none the less true for being more or less 
general Thus the kinds of cases upon which Bergson 
relies are insufficient to show the impossibility of 
prediction in the only sense in which prediction has 
practical or emotional interest. We may therefore 
leave the consideration of his arguments and address 
ourselves to the problem directly. 

The law of causation, according to which later 
events can theoretically be predicted by means of 
earlier events, has often been held to be a priori, a 
necessity of thought, a category without which science 


would be impossible. These claims seem to me 
excessive. In certain directions the law has been 
verified empirically, and in other directions there is 
no positive evidence against it. But science can use it 
where it has been found to be true, without being 
forced into any assumption as to its truth in other 
fields. We cannot, therefore, feel any a priori cer- 
tainty that causation must apply to human volitions. 

The question how far human volitions are subject to 
causal laws is a purely empirical one. Empirically it 
seems plain that the great majority of our volitions 
have causes, but it cannot, on this account, be hdd 
necessarily certain that all have causes. There axe, 
however, precisely the same kinds of reasons for 
regarding it as probable that they all have causes as 
there are in the case of physical events. 

We may suppose though this is doubtful that 
there are laws of correlation of the mental and the 
physical, in virtue of which, given the state of all the 
matter in the world, and therefore of all the brains and 
living organisms, the state of all the minds in the 
world could be inferred, while conversely the state of 
all the matter in the world could be inferred if the 
state of all the minds were given. It is obvious that 
there is some degree of correlation between brain and 
mind, and it is impossible to say how complete it 
may be. This, however, is not the point which I 
wish to elicit. What I wish to urge is that, even if 
we admit the most extreme claims of determinism 
and of correlation of mind and brain, still the conse- 
quences inimical to what is worth preserving in free 
will do not follow. The belief that they follow results, 
I think, entirely from the assimilation of causes to 
volitions, and from the notion that causes compel 
their effects in some sense analogous to that in which 


a human authority can compel a man to do what he 
would rather not do. This assimilation, as soon as 
the true nature of scientific causal laws is realized, is 
seen to be a sheer mistake. But this brings us to the 
second of the two questions which we raised in regard 
to free will, namely whether, assuming determinism, 
our actions can be in any proper sense regarded as 
compelled by outside forces. 

(2) Are human actions subject to an external com- 
pulsion t We have, in deliberation, a subjective sense 
of freedom, which is sometimes alleged against the 
view that volitions have causes. This sense of freedom, 
however, is only a sense that we can choose which 
we please of a number of alternatives : it does not 
show us that there is no causal connection between 
what we please to chose and our previous history. 
The supposed inconsistency of these two springs from 
the habit of conceiving causes as analogous to volitions 
a habit which often survives unconsciously in those 
who intend to conceive causes in a more scientific 
manner. If a cause is analogous to a volition, outside 
causes will be analogous to an alien will, and acts 
predictable from outside causes will be subject to 
compulsion. But this view of cause is one to which 
science lends no countenance. Causes, we have seen, 
do not compel their effects, any more than effects 
compel their causes. There is a mutual relation, so 
that either can be inferred from the other. When 
the geologist infers the past state of the earth from 
its present state, we should not say that the present 
state compels the past state to have been what it 
was ; yet it renders it necessary as a consequence of 
the data, in the only sense in which effects are rendered 
necessary by their causes. The difference which we 
feel, in this respect, between causes and effects is a 


mere confusion due to the fact that we remember past 
events but do not happen to have memory of the 

The apparent indeterminateness of the future, upon 
which some advocates of free will rely, is merely a 
result of our ignorance. It is plain that no desirable 
kind of free will can be dependent simply upon our 
ignorance ; for if that were the case, animals would 
be more free than men, and savages than civilized 
people. Free will in any valuable sense must be 
compatible with the fullest knowledge. Now, quite 
apart from any assumption as to causality, it is obvious 
that complete knowledge would embrace the future 
as well as the past. Our knowledge of the past is not 
wholly based upon causal inferences, but is partly 
derived from memory. It is a mere accident that we 
have no memory of the future. We might as in the 
pretended visions of seers see future events immedi- 
ately, in the way in which we see past events. They 
certainly will be what they will be, and are in this 
sense just as determined as the past. If we saw 
future events in the same immediate way in which 
we see past events, what kind of free will would still 
be possible ? Such a kind would be wholly indepen- 
dent of determinism : it could not be contrary to even 
the most entirely universal reign of causality. And 
such a kind must contain whatever is worth having in 
free will, since it is impossible to believe that mere 
ignorance can be the essential condition of any good 
thing. Let us therefore imagine a set of beings who 
know the whole future with absolute certainty, and 
let us ask ourselves whether they could have anything 
that we should call free will. 

Such beings as we are imagining would not have to 
wait for the event in order to know what decision 


they were going to adopt on some future occasion. 
They would know now what their volitions were 
going to be. But would they have any reason to 
regret this knowledge ? Surely not, unless the fore- 
seen volitions were in themselves regrettable. And it 
is less likely that the foreseen volitions would be 
regrettable if the steps which would lead to them 
were also foreseen. It is difficult not to suppose that 
what is foreseen is fated, and must happen however 
much it may be dreaded. But human actions are the 
outcome of desire, and no foreseeing can be true 
unless it takes account of desire. A foreseen volition 
will have to be one which does not become odious 
through being foreseen. The beings we are imagining 
would easily come to know the causal connections of 
volitions, and therefore their volitions would be better 
calculated to satisfy their desires than ours are. Since 
volitions are the outcome of desires, a prevision of 
volitions contrary to desires could not- be a true one. 
It must be remembered that the supposed prevision 
would not create the future any more thaa memory 
creates the past. We do not think we were necessarily 
not free in the past, merely because we can now 
remember our past volitions. Similarly, we might be 
free in the future, even if we could now see what our 
future volitions were going to be. Freedom, in short, 
in any valuable sense, demands only that our volitions 
shall be, as they are, the result of our own desires, not 
of an outside force compelling us to will what we would 
rather not will. Everything else is confusion of 
thought, due to the feeling that knowledge compels 
the happening of what it knows when this is future, 
though it is at once obvious that knowledge has no 
such power in regard to the past. Free will, therefore, 
is true in the only form which is important ; and the 


desire for other forms is a mere effect of insufficient 

What has been said on philosophical method in the 
foregoing lectures has been rather by means of illus- 
trations in particular cases than by means of general 
precepts. Nothing of any value can be said on 
method except through examples; but now, at the 
end of our course, we may collect certain general 
maxims which may possibly be a help in acquiring a 
philosophical habit of mind and a guide in looking for 
solutions of philosophic problems. 

Philosophy does not become scientific by making 
use of other sciences, in the kind of way in which, e.g. 
Herbert Spencer does. Philosophy aims at what is 
general, and the special sciences, however they may 
suggest large generalizations, cannot make them certain. 
And a hasty generalization, such as Spencer's general- 
ization of evolution, is none the less hasty because 
what is generalized is the latest scientific theory. 
Philosophy is a study apart from the other sciences : 
its results cannot be established by the other sciences, 
and conversely must not be such as some other science 
might conceivably contradict. Prophecies as to the 
future of the universe, for example, are not the business 
of philosophy; whether the universe is progressive, 
retrograde, or stationary, it is not for the philosopher 
to say. 

In order to become a scientific philosopher, a certain 
peculiar mental discipline is required. There must be 
present, first of aU, the desire to know philosophical 
truth, and this desire must be sufficiently strong to 
survive through years when there seems no hope of 
its finding any satisfaction. The desire to know 
philosophical truth is very rare in its purity, it is 


not often found even among philosophers. It is 
obscured sometimes particularly after long periods of 
fruitless search by the desire to think we know. 
Some plausible opinion presents itself, and by turning 
our attention away from the objections to it, or merely 
by not making great efforts to find objections to it, 
we may obtain the comfort of believing it, although, 
if we had resisted the wish for comfort, we should 
have come to see that the opinion was false. Again 
the desire for unadulterated truth is often obscured, 
in professional philosophers, by love of system : the 
one little fact which will not come inside the philoso- 
pher's edifice has to be pushed and tortured until it 
seems to consent. Yet the one little fact is more 
likely to be important for the future than the system 
with which it is inconsistent. Pythagoras invented a 
system which fitted admirably with all the facts he 
knew, except the incommensurability of the diagonal 
of a square and the side ; this one little fact stood out, 
and remained a fact even after Hippasos of Metapon- 
tion was drowned for revealing it. To us, the discovery 
of this fact is the chief claim of Pythagoras to immor- 
tality, while his system has become a matter of merely 
historical curiosity. 1 Love of system, therefore, and 
the system-maker's vanity which becomes associated 
with it, are among the snares that the student of 
philosophy must guard against. 

The desire to establish this or that result, or generally 
to discover evidence for agreeable results, of whatever 
kind, has of course been the chief obstacle to honest 
philosophizing. So strangely perverted do men become 
by unrecognized passions, that a determination in 

'The above remarks, for purposes of illustration, adopt 
one of several possible opinions on each of several disputed 



advance to arrive at this or that conclusion is generally 
regarded as a mark of virtue, and those whose studies 
lead to an opposite conclusion are thought to be wicked. 
No doubt it is commoner to wish to arrive at an agree- 
able result than to wish to arrive at a true result. 
But only those in whom the desire to arrive at a true 
result is paramount can hope to serve any good purpose 
by the study of philosophy. 

But even when the desire to know exists in the requisite 
strength, the mental vision by which abstract truth 
is recognized is hard to distinguish from vivid imagin- 
ability and consonance with mental habits. It is 
necessary to practise methodological doubt, like 
Descartes, in order to loosen the hold of mental habits ; 
and it is necessary to cultivate logical imagination, in 
order to have a number of hypotheses at command, 
and not to be the slave of the one which common 
sense has rendered easy to imagine. These two 
processes, of doubting the familiar and imagining the 
unfamiliar, are correlative, and form the chief part of 
the mental training required for a philosopher. 

The naive beliefs which we find in ourselves when 
we first begin the process of philosophic reflection 
may turn out, in the end, to be almost all capable of 
a true interpretation; but they ought all, before 
being admitted into philosophy, to undergo the ordeal 
of sceptical criticism. Until they have gone through 
this ordeal, they are mere blind habits, ways of be- 
having rather than intellectual convictions. And 
although it may be that a majority will pass the 
test, we may be pretty sure that some will not, and 
that a serious readjustment of our outlook ought to 
result. In order to break the dominion of habit, we 
must do our best to doubt the senses, reason, morals, 
everything in short. In some directions, doubt will 


be found possible ; in others, it will be checked by 
that direct vision of abstract truth upon which the 
possibility of philosophical knowledge depends. 

At the same time, and as an essential aid to the 
direct perception of the truth, it is necessary to acquire 
fertility in imagining abstract hypotheses. This is, 
I think, what has most of all been lacking hitherto in 
philosophy. So meagre was the logical apparatus 
that all the hypotheses philosophers could imagine 
were found to be inconsistent with the facts. Too 
often this state of things led to the adoption of heroic 
measures, such as a wholesale denial of the facts, 
when an imagination better stocked with logical tools 
would have found a key to unlock the mystery. It is 
in this way that the study of logic becomes the central 
study in philosophy : it gives the method of research 
in philosophy, just as mathematics gives the method 
in physics. And as physics, which, from Plato to the 
Renaissance, was as unprogressive, dim, and supersti- 
tious as philosophy, became a science through Galileo's 
fresh observation of facts and subsequent mathematical 
manipulation, so philosophy, in our own day, is 
becoming scientific through the simultaneous acquisi- 
tion of new facts and logical methods. 

In spite, however, of the new possibility of progress 
in philosophy, the first effect, as in the case of physics, 
is to diminish very greatly the extent of what is 
thought to be known. Before Galileo, people believed 
themselves possessed of immense knowledge on all 
the most interesting questions in physics. He estab- 
lished certain facts as to the way in which bodies fall, 
not very interesting on their own account, but of 
quite immeasurable interest as examples of real 
knowledge and of a new method whose future fruitful- 
ness he himself divined. But his few facts sufficed to 


destroy the whole vast system of supposed knowledge 
handed down from Aristotle, as even the palest 
morning sun suffices to extinguish the stars. So in 
philosophy: though some have believed one system, 
and others another, almost all have been of opinion 
that a great deal was known ; but all this supposed 
knowledge in the traditional systems must be swept 
away, and a new beginning must be made, which we 
shall esteem fortunate indeed if it can attain results 
comparable to Galileo's law of falling bodies. 

By the practice of methodological doubt, if it is 
genuine and prolonged, a certain humility as to our 
knowledge is induced: we become glad to know 
anything in philosophy, however seemingly trivial. 
Philosophy has suffered from the lack of this kind of 
modesty. It has made the mistake of attacking the 
interesting problems at once, instead of proceeding 
patiently and slowly, accumulating whatever solid 
knowledge was obtainable, and trusting the great 
problems to the future. Men of science are not 
ashamed of what is intrinsically trivial, if its conse- 
quences are likely to be important; the immediate 
outcome of an experiment is hardly ever interesting 
on its own account. So in philosophy, it is often 
desirable to expend time and care on matters which, 
judged alone, might seem f rivolous, for it is often only 
through the consideration of such matters that the 
greater problems can be approached. 

When our problem has been selected, and the 
necessary mental discipline has been acquired, the 
method to be pursued is fairly uniform. The big 
problems which provoke philosophical inquiry are 
found, on examination, to be complex, and to depend 
upon a number of component problems, usually more 
abstract than those of which they are the components. 


It will generally be found that all our initial data, all 
the facts that we seem to know to begin with, suffer 
from vagueness, confusion, and complexity. Current 
philosophical ideas share these defects ; it is therefore 
necessary to create an apparatus of precise conceptions 
as general and as free from complexity as possible, 
before the data can be analysed into the kind of 
premisses which philosophy aims at discovering. In 
this process of analysis, the source of difficulty is 
tracked further and further back, growing at each 
stage more abstract, more refined, more difficult to 
apprehend. Usually it will be found that a number 
of these Artra.nHiTifl.ri1y abstract questions underlie 
any one of the big obvious problems. When every- 
thing has been done that can be done by method, a 
stage is reached where only direct philosophic vision 
can carry matters further. Here only genius will 
avail. What is wanted, as a rule, is some new effort 
of logical imagination, some glimpse of a possibility 
never conceived before, and then the direct perception 
that this possibility is realized in the case in question. 
Failure to thinfe of the right possibility leaves insoluble 
difficulties, balanced arguments pro and con, utter 
bewilderment and despair. But the right possibility, 
as a rule, when once conceived, justifies itself swiftly 
by its astonishing power of absorbing apparently 
conflicting facts. From this point onward, the work 
of the philosopher is synthetic and comparatively easy ; 
it is in the very last stage of the analysis that the real 
difficulty consists. 

Of the prospect of progress in philosophy, it would 
be rash to speak with confidence. Many of the 
traditional problems of philosophy, perhaps most of 
those which have interested a wider circle than that 
of technical students, do not appear to be soluble by 


scientific methods. Just as astronomy lost much of 
its human interest when it ceased to be astrology, so 
philosophy must lose in attractiveness as it grows 
less prodigal of promises. But to the large and still 
growing body of men engaged in the pursuit of science 
men who hitherto, not without justification, have 
turned aside from philosophy with a certain contempt 
the new method, successful already in such time- 
honoured problems as number, infinity, continuity, 
space and time, should make an appeal which the 
older methods have wholly failed to make. Physics, 
with its principle of relativity and its revolutionary 
investigations into the nature of matter, is feeling the 
need for that kind of novelty in fundamental hypo- 
theses which scientific philosophy aims at facilitating. 
The one and only condition, I believe, which is neces- 
sary in order to secure for philosophy in the near future 
an achievement surpassing all that has hitherto been 
accomplished by philosophers, is the creation of a 
school of men with scientific training and philosophical 
interests, unhampered by the traditions of the past, 
and not misled by the literary methods of those who 
copy the ancients in all except their merits. 


Absolute, 16, 48. 
Abstraction, principle of, 51, 

132 ff. 
Achilles, Zeno's argument of, 


Acquaintance, 35, 151. 
Activity, 228 ff. 
Allman, 16571. 
Analysis, 189, 209, 214, 245. 
legitimacy of, 156. 

An gnrimflnder, 13. 

Antinomies, Kant's, 159 ff. 

Aquinas, 20. 

Aristotle, 49, 164 n., 1652., 

Arrow, Zeno's argument of, 


Assertion, 61. 
Atomism, logical, 14- 
Atomists, 164. 

Belief, 67. 
primitive and derivative, 

75 ff. 
Bergson, 14, 21, 23, 29 ff., 143, 

144, 157, 162, 169, 179, 

184, 233 ff. 

Berkeley, 63, 64, 102. 
Bolzano, 169. 
Boole, 50. 

Bradley, 16, 48, 170. 
Broad, 131. 17? 
Brochard, 174 ** 
Burnet, 29 n., 164*., i66., 

175 i-> 

Calderon, 103. 

Cantor, Georg, 8, 9, 159, 169, 

I94 199. 204. 
Cantor, Moritz, 173 n. 
Categories, 48. 
Causal laws, 115, 215 ff. 

evidence for, 219 ff. 

in psychology, 222. 
Causation, 43 ff., 86, 215 ff. 

law of, 224. 

not a priori, 226, 235. 
Cause, 223, 226. 
Certainty, degrees of, 74, 75, 

Change, demands analysis, 


Cinematograph, 154, 179* 
Classes, 206. 

non-existence of, 2ioff. 
Classical tradition, 14 ff., 68. 
Complexity, 152, 162 ff. 
Compulsion, 232, 2365. 
Congruence, 199. 
Consecutiveness, 140. 
Conservation, no. 
Constituents of facts, 60, 150. 
Construction v. inference, 8. 
Contemporaries, initial, 125. 
Continuity, 70, 135 ff., 147 ff" 

159 ff- 

of change, in, 113. 136 ff. 
Correlation of mental and 

physical, 235. 
Counting, 169, 187. 192 ff- 

Couturat, 49 

Dante, 20. 


Darwin, 14, 22, 33, 41. 
Data, 72 ff., 215. 

"hard "and "soft," 77 ff. 
Dates, 123. 
Definition, 209. 
Descartes, 15, 80, 242. 
Descriptions, 206, 217. 
Desire, 231, 237. 
Determinism, 237. 
Doubt, 240. 
Dreams, 93, zoi. 
Duration, 153, 157. 

Earlier and later, 121. 
Eddington, 131 n. 
Effect, 224. 
Eleatics, 30. 
Empiricism, 46, 225. 
Enclosure, 120 ff., 127. 
Enumeration, 207. 
Euclid, 168. 
Evellin, 173. 
Evolutionism, 14, 21 ff. 
Extension, 152, 155. 
External world, knowledge of, 
70 ff. 

Fact, 60. 

atomic, 61. 
Finalism, 23. 

Form, logical, 50 ff., 190, 212. 
Fractions, 138, 184. 
Free will, 215, 231 ff. 
Frege, 15, 50, 2045. 

Galileo, 14, 69, 196, 199, 243, 


Gaye, 173 n., 180, 182. 
Geometry, 15. 
Giles, 2ii n. 
Greater and less, 199. 

Harvard, 14. 

Hegel, 13, 47 ff., 56, 173. 
" Here," 80, 99- 
Hereditary properties, 220. 
Hippasos, 168, 241. 
Hui Tzft, 21 x. 
Hume, 220, 225. 
Hypotheses in philosophy, 

Illusions, 93. 
Incommensurables, 166 ff., 

Independence, 80, 81. 

causal and logical, 81, 82. 
Indiscernibility, 147, 154. 
Indivisibles, 165. 
Induction, 43, 225. 

mathematical, 199 ff. 
Inductiveness, 194, 199 ff. 
Inference, 53, 63. 
Infinite, 8, 71, 139, 155. 

historically considered, 

159 & 

"true," 184, 185. 

positive theory of, 189 ff. 
Infinitesimals, 139. 
Instants, 122 ff., 135, 153, 219. 

defined, 124. 

Instinct v. Reason, 30 ff. 
Intellect, 32 ff. 

how displayed by friends, 

inadequacy of display, 103. 
Interpretation, 151. 

James, 14, 20, 23. 
Jourdain, i6gn. 
Jowett, 172. 
Judgment, 67. 



Kant, 13, 118, 122, 159 ff., 


Keynes, 225 . 
Knowledge about, 151. 

Language, bad, 89, 142. 
Laplace, 22. 
Laws of nature, 219 ff 
Leibniz, 23, 49, 94, 190, 195. 
Logic, 205. 

analytic not constructive, 1 8. 

Aristotelian, 15. 

and fact, 62. 

inductive, 43, 225. 

mathematical, 8, 49 ff . 

mystical, 54. 

and philosophy, 18, 42 ff., 

Logical constants, 213. 

Mach, 131, 226. 
Macran, 48 n. 
Mathematics, 49, 66. 
Matter, 83, 106 ff. 

permanence of, 107 fL 
Measurement, 167. 
Memory, 233, 237, 239. 

deductive, 15. 

logical-analytic, 7, 74, 214, 

240 ff. 

Milhaud, 172 .> 173 n. 
Mill, 43, 204. 
Montaigne, 39. 
Motion, 136, 219. 

continuous, 139, 142. 

mathematical theory of, 


perception of, 143 ff. 
Zeno's arguments on, 172 ff. 
Mysticiam, 29, 56, 70, 103. 

Newton, 41, ^53. 
Nicod, 121. 
Nietzsche, 20, 21. 
Noa, 173. 

cardinal, 137, 190 S. 

denned, 203 ff. 

finite, 165, 193 if. 

inductive, 199. 

infinite, 183, 186, 190 fL, 


reflexive, 1943. 

Occam, 112, 153. 

One and many, 172, 174. 

Order, 137. 

Parmenides, 70, 170 ff., 183. 
Past and future, 227, 237 ff. 
Peano, 50. 

Perspectives, 94 ff., 116. 
Philoponus, 1761*. 

and ethics, 37 ff. 

and mathematics, 189 ff. 

province of, 27, 36, 189, 236. 

scientific, n, 16, 18, 29, 

240 ff. 
Physics, io6ff., 153, 243, 246. 

descriptive, 227. 

verifiability of, 88, 116. 
Place, 93. 97- 

at and front, 100. 
Plato, 14, 29, 37* 55. 7 2 

1701*., 171. 
PoincarS, 131, 148. 
Points, H9ff., I35i 1^2. 

definition of, 8, 119. 
Pragmatism, 21. 
Prantl, 179. 
Predictability, 232 ff. 
Premisses, 214. 


Probability, 45. 
Propositions! 62. 

atomic, 62. 

general, 65. 

molecular, 64. 
Pythagoras, 29, 164 ff., 241. 

Race-course, Zeno's argument 

of, 175 ff. 
Realism, new, 16. 
RefLexiveness, 194 ff. 
Relations, 54. 

asymmetrical, 57. 

Bradley's reasons against, 

external, 157. 

intransitive, 58. 

multiple, 60. 

one-one, 207. 

reality of, 59. 

symmetrical, 57, 127. 

transitive, 58, 127. 
Relativity, 109, 246. 
Repetitions, 233 ff. 
Rest, 142. 

Ritter and Preller, 166 n. 
Robertson, D. S., 164 n. 
Rousseau, 30. 
Royce, 60. 

Santayana, 55. 
Scepticism, 73, 74. 
Seeing double, 93. 
Self, 81. 
Sensation, 35, 83, 131. 

and stimulus, 145. 
Sense-data, 63, 70, 72, 82, 

116, 148, 150, 216. 
and physics, 8, 71, 88, 104, 

io6ff., 146. 

infinitely numerous ? 156, 

Sense-perception, 63. 
Series, 59. 

compact, 138, 148, 183. 

continuous, 138, 139. 
Sigwart, 191. 
Simplitius, 175 n. 
Simultaneity, 121. 
Space, 80, 96, 109, 117 ff., 

absolute and relative, 153, 


antinomies of, 159 ff . 
perception of, 75. 
of perspectives, 95 ff. 
private, 96* 97- 
of touch and sight, 85, 


Spencer, 14, 22, 240. 
Spinoza, 55, 171. 
Stadium, Zeno's argument of, 

141 n., 180 ff. 
Subject-predicate, 54. 
Synthesis, 160, 189. 

Tannery, Paul, 173 n. 

Teleology, 227. 

Testimony, 74, 79, 89, 95, 

101, 215. 
Thales, 13. 

Thing-in-itself, 83, 92. 
Things, 96 ff., noff., 216. 
Time, 107, 120 ff., 135, 159 ff., 
171, 218. 

absolute or relative, 152. 

local, 109. 

private/ 128. 

Uniformities, 220. 
Unity, organic, 19. 
Universal and particular, 



Volition, 227 ff. 

Whitehead, 8, 131, 212. 
Wittgenstein, g, 213*1. 

actual and ideal, 116. 

possible, 190. 
private, 95. 

Zeller, 178. 

Zeno, 135, 140, 142, 169 ff. 









By Bertrand Russell 

rrs SCOPES AND LIMITS D^ 8 . I8j . t 

This book is intended for the general reader, not for professional 
philosophers. It begins with a brief survey of what science professes 
to know about the universe. In this survey the attempt is to be as 
far as possible impartial and impersonal; the aim is to come as near 
as our capacities permit to describing the world as it might appear 
to an observer of miraculous perceptive powers viewing it from 
without. In science, we are concerned with what we koow rather than 
what we know. We attempt to use an order in our description which 
ignores, for the moment, the feet that we are part of the universe, 
and that any account which we can give of it depends upon its effects 
upon ourselves, and is to this extent inevitably anthropocentric. 

Bertrand Russell accordingly begins with the system of galaxies, 
and passes on, by stages, to our own galaxy, our own little solar 
system, our own tiny planet, the infinitesimal specks of life upon its 
surface, and finally, as the climax of insignificance, the bodies and 
minds of those odd beings that imagine themselves the lords of 
creation and the end of the whole vast cosmos. 

But this survey, which seems to end in the pettiness of Man and 
all his concerns, is only one side of the truth. There is another side, 
which must be brought out by a survey of a different kind. In this 
second kind of survey, the question is no longer what the universe 
is, but how we come to know whatever we do know about it. In 
this survey Man again occupies the centre, as in the Ptolemaic 
astronomy. What we know of the world we know by means of 
events in our own lives, events which, but for the power of thought, 
would remain merely private. > 

The book inquires what are our data, and what are the principles 
by means of which we make our inferences. The data from which 
these inferences proceed are private to ourselves; what we call 
1 'seeing the sun" is an event in the life of the seer, from which the 
astronomer's sun has to be inferred by a long and elaborate process. 
It is evident that, if the world were a higgledy-piggledy chaos, 
inferences of this kind would be impossible; but for casual inter- 
connectedness, what happens in one place would afford no indication 
of what has happened in another. It is the process from private 
sensation and thought to impersonal science that forms the chief 
topic of the book. The road is at times difficult, but until we have 
traversed it neither the scope nor the limitations of human knowledge 
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Many readers consider this to be Bertrand Russell's most important 
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Russell's purpose in Tower was to prove that the fundamental con- 
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